# Implicit Differentiation and the Chain Rule

by Peter G.
Tags: chain, differentiation, implicit, rule
 P: 435 Hi, I was trying to understand why the chain rule is needed to differentiate expressions implicitly. I began by analyzing the equation used by most websites I visited: e.g. x2+y2 = 10 After a lot of thinking, I got to a reasoning that satisfied me... Here it goes: From my understanding, the variable y is a function of x. This function of x is being squared. This means that we can think of f(x) as part of another function (e.g. u = g(y) = y^2). Hence, y^2 is a composite function and, thus, differentiating it would require the chain rule. However, after coming across some different type of questions I am no longer sure my train of thought is valid. For example: 6x^2+17y = 0. I have read that to differentiate 17 y with respect to x we also have to apply the chain rule. This does not fit with my original reasoning (since, to my eyes, y cannot be thought of as a composite function in this case) Can anyone please help me understand why we have to use the chain rule to differentiatie implicitly? Thank you in advance!
Mentor
P: 19,774
 Quote by Peter G. Hi, I was trying to understand why the chain rule is needed to differentiate expressions implicitly. I began by analyzing the equation used by most websites I visited: e.g. x2+y2 = 10 After a lot of thinking, I got to a reasoning that satisfied me... Here it goes: From my understanding, the variable y is a function of x. This function of x is being squared. This means that we can think of f(x) as part of another function (e.g. u = g(y) = y^2). Hence, y^2 is a composite function and, thus, differentiating it would require the chain rule. However, after coming across some different type of questions I am no longer sure my train of thought is valid. For example: 6x^2+17y = 0. I have read that to differentiate 17 y with respect to x we also have to apply the chain rule. This does not fit with my original reasoning (since, to my eyes, y cannot be thought of as a composite function in this case)
No need to apply the chain rule here. You're just differentiating y with respect to x (to get dy/dx). In your previous example, you did need to use the chain rule, since d/dx(y2) = d/dy(y2) * dy/dx. Here we had a function of y, and y itself was a function of x.
 Quote by Peter G. Can anyone please help me understand why we have to use the chain rule to differentiatie implicitly? Thank you in advance!
 P: 435 Oh, perfect, that is great news! I must have misread the solution to the last problem. Thank you very much!
P: 18

## Implicit Differentiation and the Chain Rule

 Quote by Peter G. I have read that to differentiate 17 y with respect to x we also have to apply the chain rule. This does not fit with my original reasoning (since, to my eyes, y cannot be thought of as a composite function in this case)
You can think of y as a composite function, if you think of x as a function of x.
P: 18
 Quote by Mark44 No need to apply the chain rule here. You're just differentiating y with respect to x (to get dy/dx). In your previous example, you did need to use the chain rule, since d/dx(y2) = d/dy(y2) * dy/dx. Here we had a function of y, and y itself was a function of x.
Technically you do use the chain rule, because 17y is a function of y.
P: 515
 Quote by Peter G. I have read that to differentiate 17 y with respect to x we also have to apply the chain rule.
Or by using the product rule, together with the fact that
$$\frac{\mathrm{d}(17)}{\mathrm{d}x} = 0.$$
Mentor
P: 19,774
 Quote by shortydeb Technically you do use the chain rule, because 17y is a function of y.
Just because 17y is a function of y doesn't mean that the chain rule needs to be used.
 Quote by pasmith Or by using the product rule, together with the fact that $$\frac{\mathrm{d}(17)}{\mathrm{d}x} = 0.$$
Or better yet, the constant multiple rule. d/dx(17y) = 17 * dy/dx.

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