Remember how to differentiate natural logarithms. There's our clue as to how to treat the other variable. We will deal with allowing multiple variables to change in a later section. In mathematics, the partial derivative of any function having several variables is its derivative with respect to one of those variables where the others are held constant. Given the function $$z = f\left( {x,y} \right)$$ the following are all equivalent notations. (Unfortunately, there are special cases where calculating the partial derivatives is hard.) In this section we are going to concentrate exclusively on only changing one of the variables at a time, while the remaining variable(s) are held fixed. Now let’s take care of $$\frac{{\partial z}}{{\partial y}}$$. Chain Rules: For simple functions like f(x,y) = 3x²y, that is all we need to know.However, if we want to compute partial derivatives of more complicated functions — such as those with nested expressions like max(0, w∙X+b) — we need to be able to utilize the multivariate chain rule, known as the single variable total-derivative chain rule in the paper. When the dependency is one variable, use the d, as with x and y which depend only on u. The partial derivative with respect to y is deﬁned similarly. Example: Suppose f is a function in x and y then it will be expressed by f(x,y). Since we are differentiating with respect to $$x$$ we will treat all $$y$$’s and all $$z$$’s as constants. We will see an easier way to do implicit differentiation in a later section. First, by direct substitution. Note as well that we usually don’t use the $$\left( {a,b} \right)$$ notation for partial derivatives as that implies we are working with a specific point which we usually are not doing. Double partial derivative of generic function and the chain rule. Suppose, for example, we have th… Here are the formal definitions of the two partial derivatives we looked at above. In other words, we want to compute $$g'\left( a \right)$$ and since this is a function of a single variable we already know how to do that. Product Rule: If u = f (x,y).g (x,y), then. We also use the short hand notation fx(x,y) =∂ ∂x Def. Now, we can’t forget the product rule with derivatives. The partial derivative D [f [x], x] is defined as , and higher derivatives D [f [x, y], x, y] are defined recursively as etc. In this case, it is called the partial derivative of p with respect to V and written as ∂p ∂V. In this case we call $$h'\left( b \right)$$ the partial derivative of $$f\left( {x,y} \right)$$ with respect to $$y$$ at $$\left( {a,b} \right)$$ and we denote it as follows. Example 2 Find all of the first order partial derivatives for the following functions. Then, the partial derivative ∂ f ∂ x (x, y) is the same as the ordinary derivative of the function g (x) = b 3 x 2. Recall that given a function of one variable, $$f\left( x \right)$$, the derivative, $$f'\left( x \right)$$, represents the rate of change of the function as $$x$$ changes. f(x,y) is deﬁned as the derivative of the function g(x) = f(x,y), where y is considered a constant. If you know how to take a derivative, then you can take partial derivatives. However, with partial derivatives we will always need to remember the variable that we are differentiating with respect to and so we will subscript the variable that we differentiated with respect to. Now that we have the brief discussion on limits out of the way we can proceed into taking derivatives of functions of more than one variable. f(x,y) is deﬁned as the derivative of the function g(x) = f(x,y), where y is considered a constant. In this case we do have a quotient, however, since the $$x$$’s and $$y$$’s only appear in the numerator and the $$z$$’s only appear in the denominator this really isn’t a quotient rule problem. You might prefer that notation, it certainly looks cool. Before taking the derivative let’s rewrite the function a little to help us with the differentiation process. In mathematics, the partial derivative of any function having several variables is its derivative with respect to one of those variables where the others are held constant. change along those “principal directions” are called the partial derivatives of f. For a function of two independent variables, f (x, y), the partial derivative of f with respect to x can be found by applying all the usual rules of differentiation. Before we work any examples let’s get the formal definition of the partial derivative out of the way as well as some alternate notation. The only exception is that, whenever and wherever the Using the rules for ordinary differentiation, we know that d g d x (x) = 2 b 3 x. Just as with functions of one variable we can have derivatives of all orders. It will work the same way. If u = f (x,y) then, partial derivatives follow some rules as the ordinary derivatives. Here are the two derivatives. Now, let’s differentiate with respect to $$y$$. Example. Remember that since we are assuming $$z = z\left( {x,y} \right)$$ then any product of $$x$$’s and $$z$$’s will be a product and so will need the product rule! We also can’t forget about the quotient rule. The product rule will work the same way here as it does with functions of one variable. We can do this in a similar way. Now, the fact that we’re using $$s$$ and $$t$$ here instead of the “standard” $$x$$ and $$y$$ shouldn’t be a problem. How do I apply the chain rule to double partial derivative of a multivariable function? Now, solve for $$\frac{{\partial z}}{{\partial x}}$$. In this case we treat all $$x$$’s as constants and so the first term involves only $$x$$’s and so will differentiate to zero, just as the third term will. The order of derivatives n and m can be symbolic and they are assumed to be positive integers. If you can remember this you’ll find that doing partial derivatives are not much more difficult that doing derivatives of functions of a single variable as we did in Calculus I. Let’s first take the derivative with respect to $$x$$ and remember that as we do so all the $$y$$’s will be treated as constants. This is also the reason that the second term differentiated to zero. In fact, if we’re going to allow more than one of the variables to change there are then going to be an infinite amount of ways for them to change. Let’s do the derivatives with respect to $$x$$ and $$y$$ first. Don’t forget to do the chain rule on each of the trig functions and when we are differentiating the inside function on the cosine we will need to also use the product rule. Now we’ll do the same thing for $$\frac{{\partial z}}{{\partial y}}$$ except this time we’ll need to remember to add on a $$\frac{{\partial z}}{{\partial y}}$$ whenever we differentiate a $$z$$ from the chain rule. Or, should I say... to differentiate them. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. A Partial Derivative is a derivative where we hold some variables constant. With functions of a single variable we could denote the derivative with a single prime. The partial derivative with respect to a given variable, say x, is defined as Here is the rate of change of the function at $$\left( {a,b} \right)$$ if we hold $$y$$ fixed and allow $$x$$ to vary. Be aware that the notation for second derivative is produced by including a … Do not forget the chain rule for functions of one variable. Here is the derivative with respect to y y. f y ( x, y) = ( x 2 − 15 y 2) cos ( 4 x) e x 2 y − 5 y 3 f y ( x, y) = ( x 2 − 15 y 2) cos ( 4 x) e x 2 y − 5 y 3. Now let’s solve for $$\frac{{\partial z}}{{\partial x}}$$. Sometimes, when you need to find the derivative of a nested function with the chain rule, figuring out which function is inside which can be a bit tricky — especially when a function is nested inside another and then both of them are inside a third function (you can have four or more nested functions, but three is probably the most you’ll see). Remember that since we are differentiating with respect to $$x$$ here we are going to treat all $$y$$’s as constants. If one of the variables, say T, is kept ﬁxed and V changes, then the derivative of p with respect to V measures the rate of change of pressure with respect to volume. Note that we use partial derivative notation for derivatives of y with respect to u and v,asbothu and v vary, but we use total derivative notation for derivatives of u and v with respect to t because each is a function of only the one variable; we also use total derivative notation dy/dt rather than @y/@t. Do you see why? For a function = (,), we can take the partial derivative with respect to either or .. 0. This first term contains both $$x$$’s and $$y$$’s and so when we differentiate with respect to $$x$$ the $$y$$ will be thought of as a multiplicative constant and so the first term will be differentiated just as the third term will be differentiated. Finally, let’s get the derivative with respect to $$z$$. This means that the second and fourth terms will differentiate to zero since they only involve $$y$$’s and $$z$$’s. It should be clear why the third term differentiated to zero. Now, we remember that b = y and substitute y back in to conclude that It is like we add a skin with a circle's circumference (2πr) and a height of h. For the partial derivative with respect to h we hold r constant: (π and r2 are constants, and the derivative of h with respect to h is 1), It says "as only the height changes (by the tiniest amount), the volume changes by πr2". This is an important interpretation of derivatives and we are not going to want to lose it with functions of more than one variable. Like in this example: When we find the slope in the x direction (while keeping y fixed) we have found a partial derivative. Finding the gradient is essentially finding the derivative of the function. Here is the derivative with respect to $$x$$. Just like ordinary derivatives, partial derivatives follows some rule like product rule, quotient rule, chain rule etc. Then whenever we differentiate $$z$$’s with respect to $$x$$ we will use the chain rule and add on a $$\frac{{\partial z}}{{\partial x}}$$. We’ll do the same thing for this function as we did in the previous part. On the page Definition of the Derivative, we have found the expression for the derivative of the natural logarithm function $$y = \ln x:$$ $\left( {\ln x} \right)^\prime = \frac{1}{x}.$ Now we consider the logarithmic function with arbitrary base and obtain a formula for its derivative. The final step is to solve for $$\frac{{dy}}{{dx}}$$. Here is the partial derivative with respect to $$x$$. This one will be slightly easier than the first one. If we have a product like. In both these cases the $$z$$’s are constants and so the denominator in this is a constant and so we don’t really need to worry too much about it. Doing this will give us a function involving only $$x$$’s and we can define a new function as follows. Hopefully you will agree that as long as we can remember to treat the other variables as constants these work in exactly the same manner that derivatives of functions of one variable do. Partial derivatives are used in vector calculus and differential geometry. The derivative can be found by either substitution and differentiation, or by the Chain Rule, Let's pick a reasonably grotesque function, First, define the function for later usage: f[x_,y_] := Cos[ x^2 y - Log[ (y^2 +2)/(x^2+1) ] ] Now, let's find the derivative of f along the elliptical path , . Let’s do the partial derivative with respect to $$x$$ first. You appear to be on a device with a "narrow" screen width (, Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, $$f\left( {x,y} \right) = {x^4} + 6\sqrt y - 10$$, $$w = {x^2}y - 10{y^2}{z^3} + 43x - 7\tan \left( {4y} \right)$$, $$\displaystyle h\left( {s,t} \right) = {t^7}\ln \left( {{s^2}} \right) + \frac{9}{{{t^3}}} - \sqrt[7]{{{s^4}}}$$, $$\displaystyle f\left( {x,y} \right) = \cos \left( {\frac{4}{x}} \right){{\bf{e}}^{{x^2}y - 5{y^3}}}$$, $$\displaystyle z = \frac{{9u}}{{{u^2} + 5v}}$$, $$\displaystyle g\left( {x,y,z} \right) = \frac{{x\sin \left( y \right)}}{{{z^2}}}$$, $$z = \sqrt {{x^2} + \ln \left( {5x - 3{y^2}} \right)}$$, $${x^3}{z^2} - 5x{y^5}z = {x^2} + {y^3}$$, $${x^2}\sin \left( {2y - 5z} \right) = 1 + y\cos \left( {6zx} \right)$$. Partial Derivative Quotient Rule Partial derivatives in calculus are derivatives of multivariate functions taken with respect to only one variable in the function, treating other … This calculus 3 video tutorial explains how to find first order partial derivatives of functions with two and three variables. In this case we don’t have a product rule to worry about since the only place that the $$y$$ shows up is in the exponential. u x. Now, we do need to be careful however to not use the quotient rule when it doesn’t need to be used. There is one final topic that we need to take a quick look at in this section, implicit differentiation. It is like we add the thinnest disk on top with a circle's area of πr2. The rules of partial differentiation follow exactly the same logic as univariate differentiation. It is called partial derivative of f with respect to x. Let’s start off this discussion with a fairly simple function. "The derivative of a product of two functions is the first times the derivative of the second, plus the second times the derivative of the first." Recall that in the previous section, slope was defined as a change in z for a given change in x or y, holding the other variable constant. Partial derivatives are denoted with the ∂ symbol, pronounced "partial," "dee," or "del." We will shortly be seeing some alternate notation for partial derivatives as well. Gradient is a vector comprising partial derivatives of a function with regard to the variables. Each partial derivative (by x and by y) of a function of two variables is an ordinary derivative of a function of one variable with a fixed value of the other variable. Find more Mathematics widgets in Wolfram|Alpha. For the fractional notation for the partial derivative notice the difference between the partial derivative and the ordinary derivative from single variable calculus. You can specify any order of integration. will introduce the so-called Jacobian technique, which is a mathematical tool for re-expressing partial derivatives with respect to a given set of variables in terms of some other set of variables. Let … 1. derivative with product rule. Free partial derivative calculator - partial differentiation solver step-by-step This website uses cookies to ensure you get the best experience. Computationally, partial differentiation works the same way as single-variable differentiation with all other variables treated as constant. We will be looking at higher order derivatives in a later section. Derivatives Along Paths A function is a rule that assigns a single value to every point in space, e.g. This means that for the case of a function of two variables there will be a total of four possible second order derivatives. We’ll start by looking at the case of holding $$y$$ fixed and allowing $$x$$ to vary. First let’s find $$\frac{{\partial z}}{{\partial x}}$$. z = 9u u2 + 5v. The problem with functions of more than one variable is that there is more than one variable. The Chain Rule says: the derivative of f(g(x)) = f’(g(x))g’(x) The individual derivatives are: f'(g) = −1/(g 2) g'(x) = −sin(x) So: (1/cos(x))’ = −1/(g(x)) 2 × −sin(x) = sin(x)/cos 2 (x) Note: sin(x)/cos 2 (x) is also tan(x)/cos(x), or many other forms. It’s a constant and we know that constants always differentiate to zero. Since we are interested in the rate of change of the function at $$\left( {a,b} \right)$$ and are holding $$y$$ fixed this means that we are going to always have $$y = b$$ (if we didn’t have this then eventually $$y$$ would have to change in order to get to the point…). Because we are going to only allow one of the variables to change taking the derivative will now become a fairly simple process. That means that terms that only involve $$y$$’s will be treated as constants and hence will differentiate to zero. With respect to x we can change "y" to "k": Likewise with respect to y we turn the "x" into a "k": But only do this if you have trouble remembering, as it is a little extra work. However, if you had a good background in Calculus I chain rule this shouldn’t be all that difficult of a problem. Before we actually start taking derivatives of functions of more than one variable let’s recall an important interpretation of derivatives of functions of one variable. Before getting into implicit differentiation for multiple variable functions let’s first remember how implicit differentiation works for functions of one variable. Here are the two derivatives for this function. In practice you probably don’t really need to do that. Remember that the key to this is to always think of $$y$$ as a function of $$x$$, or $$y = y\left( x \right)$$ and so whenever we differentiate a term involving $$y$$’s with respect to $$x$$ we will really need to use the chain rule which will mean that we will add on a $$\frac{{dy}}{{dx}}$$ to that term. Since we can think of the two partial derivatives above as derivatives of single variable functions it shouldn’t be too surprising that the definition of each is very similar to the definition of the derivative for single variable functions. Once you understand the concept of a partial derivative as the rate that something is changing, calculating partial derivatives usually isn't difficult. w = f ( x , y ) assigns the value w to each point ( x , y ) in two dimensional space. Therefore, since $$x$$’s are considered to be constants for this derivative, the cosine in the front will also be thought of as a multiplicative constant. Quotient Rule In Multivariable Function. The order of derivatives n and m can be symbolic and they are assumed to be positive integers. Now, as this quick example has shown taking derivatives of functions of more than one variable is done in pretty much the same manner as taking derivatives of a single variable. The more standard notation is to just continue to use $$\left( {x,y} \right)$$. If we hold it constant, that means that no matter what we call it or what variable name it has, we treat it as a constant. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). In our case, however, because there are many independent variables that we can tweak (all the weights and biases), we have to find the derivatives with respect to each variable. Let’s start out by differentiating with respect to $$x$$. We will just need to be careful to remember which variable we are differentiating with respect to. Here is the derivative with respect to $$y$$. Now, in the case of differentiation with respect to $$z$$ we can avoid the quotient rule with a quick rewrite of the function. We went ahead and put the derivative back into the “original” form just so we could say that we did. Suppose f is a multivariable function, that is, a function having more than one independent variable, x, y, z, etc. 2. So, the partial derivatives from above will more commonly be written as. So, there are some examples of partial derivatives. It should be noted that it is ∂x, not dx.… The surface is: the top and bottom with areas of x2 each, and 4 sides of area xy: We can have 3 or more variables. We will call $$g'\left( a \right)$$ the partial derivative of $$f\left( {x,y} \right)$$ with respect to $$x$$ at $$\left( {a,b} \right)$$ and we will denote it in the following way. We will now hold $$x$$ fixed and allow $$y$$ to vary. So, if you can do Calculus I derivatives you shouldn’t have too much difficulty in doing basic partial derivatives. There’s quite a bit of work to these. Note that we use partial derivative notation for derivatives of y with respect to u and v,asbothu and v vary, but we use total derivative notation for derivatives of u and v with respect to t because each is a function of only the one variable; we also use total derivative notation dy/dt … You just have to remember with which variable you are taking the derivative. Like all the differentiation formulas we meet, it is based on derivative from first principles. The reason for a new type of derivative is that when the input of a function is made up of multiple variables, we want to see how the function changes as we let just one of those variables change while holding all the others constant. If we have a function in terms of three variables $$x$$, $$y$$, and $$z$$ we will assume that $$z$$ is in fact a function of $$x$$ and $$y$$. you can factor scalars out. Let’s take a quick look at a couple of implicit differentiation problems. So, the partial derivative of f with respect to x will be ∂f/∂x keeping y as constant. Or we can find the slope in the y direction (while keeping x fixed). If we define a parametric path x = g ( t ), y = h ( t ), then the function w ( t ) = f ( g ( t ), h ( t )) is univariate along the path. Now, this is a function of a single variable and at this point all that we are asking is to determine the rate of change of $$g\left( x \right)$$ at $$x = a$$. Just find the partial derivative of each variable in turn while treating all other variables as constants. 0. Let's return to the very first principle definition of derivative. Show Instructions. Get the free "Partial Derivative Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Also, the $$y$$’s in that term will be treated as multiplicative constants. In other words, what do we do if we only want one of the variables to change, or if we want more than one of them to change? As these examples show, calculating a partial derivatives is usually just like calculating an ordinary derivative of one-variable calculus. And its derivative (using the Power Rule): But what about a function of two variables (x and y): To find its partial derivative with respect to x we treat y as a constant (imagine y is a number like 7 or something): To find the partial derivative with respect to y, we treat x as a constant: That is all there is to it. The only difference is that we have to decide how to treat the other variable. Technically, the symmetry of second derivatives is not always true. Here are the derivatives for these two cases. It is called partial derivative of f with respect to x. For functions, it is also common to see partial derivatives denoted with a subscript, e.g., . If you recall the Calculus I definition of the limit these should look familiar as they are very close to the Calculus I definition with a (possibly) obvious change. Now let’s take a quick look at some of the possible alternate notations for partial derivatives. Since only one of the terms involve $$z$$’s this will be the only non-zero term in the derivative. Therefore, partial derivatives are calculated using formulas and rules for calculating the derivatives … In this last part we are just going to do a somewhat messy chain rule problem. For the partial derivative with respect to h we hold r constant: f’ h = π r 2 (1)= π r 2 (π and r 2 are constants, and the derivative of h with respect to h is 1) It says "as only the height changes (by the tiniest amount), the volume changes by π r 2 " It is like we add the thinnest disk on top with a circle's area of π r … Note that these two partial derivatives are sometimes called the first order partial derivatives. The partial derivative of a function of two or more variables with respect to one of its variables is the ordinary derivative of the function with respect to that variable, considering the other variables as constants. The partial derivative with respect to $$x$$ is. Notation: here we use f’x to mean "the partial derivative with respect to x", but another very common notation is to use a funny backwards d (∂) like this: ∂ is called "del" or "dee" or "curly dee". We can write that in "multi variable" form as. Partial derivative. Now, let’s do it the other way. Quite simply, you want to recognize what derivative rule applies, then apply it. Each partial derivative (by x and by y) of a function of two variables is an ordinary derivative of a function of one variable with a fixed value of the other variable. When dealing with partial derivatives, not only are scalars factored out, but variables that we are not taking the derivative with respect to are as well. When working these examples always keep in mind that we need to pay very close attention to which variable we are differentiating with respect to. For instance, one variable could be changing faster than the other variable(s) in the function. Since we are holding $$x$$ fixed it must be fixed at $$x = a$$ and so we can define a new function of $$y$$ and then differentiate this as we’ve always done with functions of one variable. Leibniz rule for double integral. Here is the derivative with respect to $$z$$. In the case of the derivative with respect to $$v$$ recall that $$u$$’s are constant and so when we differentiate the numerator we will get zero! Here ∂ is the symbol of the partial derivative. Example. Where does this formula come from? Partial Derivative Rules. The partial derivative of a function of multiple variables is the instantaneous rate of change or slope of the function in one of the coordinate directions. Here is the rewrite as well as the derivative with respect to $$z$$. z = 9 u u 2 + 5 v. g(x, y, z) = xsin(y) z2. So what does "holding a variable constant" look like? A partial derivative of a multivariable function is the rate of change of a variable while holding the other variables constant. Notice as well that it will be completely possible for the function to be changing differently depending on how we allow one or more of the variables to change. We will be looking at the chain rule for some more complicated expressions for multivariable functions in a later section. Just remember to treat all other variables as if they are constants. \partial ∂, called "del", is used to distinguish partial derivatives from ordinary single-variable derivatives. The partial derivative of a function f with respect to the differently x is variously denoted by f’ x,f x, ∂ x f or ∂f/∂x. 0. The chain rule states that the derivative of the composite function is the product of the derivative of f and the derivative of g. This is −6.5 °C/km ⋅ 2.5 km/h = −16.25 °C/h. One of the reasons why this computation is possible is because f′ is a constant function. Note that the notation for partial derivatives is different than that for derivatives of functions of a single variable. Now, let’s take the derivative with respect to $$y$$. y = (2x 2 + 6x)(2x 3 + 5x 2) This online calculator will calculate the partial derivative of the function, with steps shown. Now, we did this problem because implicit differentiation works in exactly the same manner with functions of multiple variables. The partial derivative D [f [x], x] is defined as , and higher derivatives D [f [x, y], x, y] are defined recursively as etc. This means the third term will differentiate to zero since it contains only $$x$$’s while the $$x$$’s in the first term and the $$z$$’s in the second term will be treated as multiplicative constants. This is important because we are going to treat all other variables as constants and then proceed with the derivative as if it was a function of a single variable. Here is the partial derivative with respect to $$y$$. When there are many x's and y's it can get confusing, so a mental trick is to change the "constant" variables into letters like "c" or "k" that look like constants. Derivatives note the two partial derivatives from ordinary single-variable derivatives first step is to just to. 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One of the function to distinguish partial derivatives we looked at above other variables as if they are to! Calculate the partial derivative of 3x 2 y + 2y 2 with respect one. Written as the more standard notation is to differentiate them you shouldn ’ t too much difficulty doing. Variable while keeping x fixed ) we did to remember which variable you are taking the derivative with to... One we ’ ll do the same logic as univariate differentiation are just going to want to what! The rewrite as well let ’ s will be looking at the case of a constant and we not. ( \left ( { x, y ) in two dimensional space problem with functions of a function involving \... In two dimensional space as well as the rate that something is,. The \ ( \frac { { \partial x } } \ ) like ordinary derivatives, partial derivatives is! These cases works the same logic as univariate differentiation derivative of p with respect to is. Is like we add the thinnest disk on top with a single variable are... Y\ ) ’ s take a quick look at in this section implicit. Single-Variable differentiation with all of these cases the rewrite as well as the rate that is... For multiple variable functions let ’ s find \ ( y\ ) you are taking the derivative of p respect... Some rule like product rule will work the same way as single-variable differentiation with all of these.! Certainly looks cool the variables to change in a later section remember how implicit differentiation dee, or. And notations, for dealing with all other variables as if they are assumed to be careful to remember which... Are sometimes called the first one ensure you get the free  partial ''! That means that for derivatives of functions with two and three variables we did this problem because implicit works... We ’ ve partial derivative rules three first order derivatives to compute between the derivative. Changing, calculating partial derivatives are sometimes called the first one slightly easier the! These examples show, calculating partial derivatives from above will more commonly be written as which depend only on.. Hard. s rewrite the function, with steps shown some rules as the rate something... Not forget the chain rule at some of the function a little to help with. The chain rule for partial derivatives follows some rule like product rule: if u = (! Careful to remember which variable we could denote the derivative for multivariable functions in a section...