Implicit Differentiation and the Chain Rule

[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)

Can anyone please help me understand why we have to use the chain rule to differentiatie implicitly?

Thank you in advance!

 

[Mark44]

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.

Can anyone please help me understand why we have to use the chain rule to differentiatie implicitly?

Thank you in advance!

 

[Peter G.]

Oh, perfect, that is great news! I must have misread the solution to the last problem. Thank you very much!

 

[shortydeb]

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.

 

[shortydeb]

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.

 

[pasmith]

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
<br /> \frac{\mathrm{d}(17)}{\mathrm{d}x} = 0.<br />

 

[Mark44]

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.

Or by using the product rule, together with the fact that
<br /> \frac{\mathrm{d}(17)}{\mathrm{d}x} = 0.<br />

Or better yet, the constant multiple rule. d/dx(17y) = 17 * dy/dx.