Trouble With Understanding Implicit Differentiation

• nicksbyman
In summary, implicit differentiation is used when the variable of the function we are differentiating disagrees with the denominator of d/dx. When the variables agree, explicit differentiation can be used. The chain rule is used in implicit differentiation because we are differentiating a function of x with respect to x, so we keep the dy/dx term. When differentiating with respect to y, the simple power rule applies. This is because the derivative of the inside of the function is 1, so we do not need to write it. Implicit differentiation can also be thought of as a composition of
nicksbyman
I don't think I fully understand implicit differentiation. I have read my textbook and watched many videos, and I think I will get an A on my test on this solely by memorizing the rules, but I would really like to understand this topic. From what I know, you are supposed to use implicit differentiation when the variable of the thing you are taking the derivative of disagrees with the denominator of d/dx and when they agree use explicit differentiation. I understand that there is a problem with taking the derivative of an expression when the variables disagree, but I don't understand why you can use the chain rule when the variables disagree but you cannot use the simple power rule (i.e. d/dx [yˆ2] = 2y(dy/dx) ≠ 2y). It seems arbitrary; why are you allowed to use the chain rule and not the simple power rule?

Thanks

the difference is you are differentiating y = y(x), a function of x, with respect to x, so you keep the dy/dx term

if you were differentiating with respect to y then the power law applies
$$\frac{d}{dy}y^2 = 2y$$

and in fact you are using both the chain rule and the power rule

^I know, but why? Why do you only use the chain rule when the variables disagree?

nicksbyman said:
^I know, but why? Why do you only use the chain rule when the variables disagree?

I think you actually use the chain rule when the variable agrees. It's just that the derivative of the inside of say X^3 is 1, (that is, the derivative with respect to x of x). So you have something like this:
$$\frac{d}{dx}x^{3}=3x^{2}(\frac{d}{dx}x)$$
The latter part is just one, so we don't have to write it.

Think of it like a composition:
$$f(x) = x^{3}$$
$$g(x) = x$$
$$(f \circ g)(x) = f(g(x)) = (x)^3$$

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yeah so just to add to quarkcharmers post

consider the function f(x) = x

then consider g(f) = f^2

taking teh composition of functions we have
$$g(f(x)) = (f(x))^2 = x^2$$

when we differntiate w.r.t. x
$$\frac{d}{dx}g(f(x)) = \frac{dg(f(x))}{dx}g(f(x))\frac{df(x)}{dx} = g'(f(x))f'(x) = 2f(x).1 = 2x$$

so the key part isfor f(x) = x
$$\frac{df(x)}{dx} = f'(x) = \frac{d(x)}{dx} = 1$$

Last edited:

1. What is implicit differentiation?

Implicit differentiation is a method used to find the derivative of a function that is not explicitly expressed in terms of the independent variable. This means that the function may contain both the independent variable and its derivative, making it difficult to find the derivative using traditional methods.

2. When is implicit differentiation used?

Implicit differentiation is typically used when trying to find the derivative of a function that is in implicit form, such as a curve or surface. It is also used when the explicit form of the function is too complex to differentiate using traditional methods.

3. How is implicit differentiation performed?

To perform implicit differentiation, the chain rule is used to differentiate each term in the function with respect to the independent variable. The derivative of the dependent variable is then isolated on one side of the equation, while all other terms are on the other side. The derivative is then solved for.

4. What are the challenges with understanding implicit differentiation?

The main challenge with understanding implicit differentiation is recognizing when it should be used and being able to apply the chain rule correctly. It also requires a strong understanding of algebra and the ability to manipulate equations to isolate the derivative.

5. Why is implicit differentiation important in science?

Implicit differentiation is important in science because it allows for the calculation of derivatives for functions that are not explicitly expressed in terms of the independent variable. This is crucial in many scientific fields, such as physics and engineering, where complex functions are common.

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