Implicit Differentiation and the Chain Rule

In summary, the chain rule is used to differentiate expressions implicitly because the variable being differentiated is a function of another variable, which itself is a function of the original variable. This creates a composite function that requires the chain rule to be applied. However, in some cases, such as when the constant 17 is involved, the chain rule may not be necessary and can be replaced by other differentiation rules.
  • #1
Peter G.
442
0
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!
 
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  • #2
Peter G. said:
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.
Peter G. said:
Can anyone please help me understand why we have to use the chain rule to differentiatie implicitly?

Thank you in advance!
 
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Likes 1 person
  • #3
Oh, perfect, that is great news! I must have misread the solution to the last problem. Thank you very much! :smile:
 
  • #4
Peter G. said:
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.
 
  • #5
Mark44 said:
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.
 
  • #6
Peter G. said:
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
[tex]
\frac{\mathrm{d}(17)}{\mathrm{d}x} = 0.
[/tex]
 
  • #7
shortydeb said:
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.
pasmith said:
Or by using the product rule, together with the fact that
[tex]
\frac{\mathrm{d}(17)}{\mathrm{d}x} = 0.
[/tex]
Or better yet, the constant multiple rule. d/dx(17y) = 17 * dy/dx.
 

1. What is implicit differentiation?

Implicit differentiation is a method used to find the derivative of a function that is not in the form of y = f(x). It is commonly used when the equation cannot be easily solved for y.

2. How is implicit differentiation different from explicit differentiation?

Explicit differentiation is used to find the derivative of a function where y is explicitly written in terms of x. Implicit differentiation is used when the equation is not explicitly solved for y, and therefore requires the use of the chain rule.

3. What is the chain rule and how is it used in implicit differentiation?

The chain rule is a formula used to find the derivative of a composite function. In implicit differentiation, the chain rule is used to find the derivative of the outer function while treating the inner function as a variable.

4. Can the chain rule be used in all cases of implicit differentiation?

No, in some cases the chain rule may not be applicable. This can happen when the equation is not in a form that allows for the use of the chain rule, or when the equation is not differentiable at certain points.

5. Why is implicit differentiation useful?

Implicit differentiation allows us to find the derivative of a function when it is not in a form that can be easily differentiated. It is a powerful tool in calculus and is used in applications such as optimization, related rates, and curve sketching.

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