Implicit Differentiation | Calculus Help

Implicit Differentiation for Calculus Problems

This very powerful differentiation process follows from the chain rule.
$$u = g(f(x)) \\
\frac{du}{dx} = g'(f(x)) \times f'(x)$$
We’ve done quite a few differentiation and derivatives, but they all have been differentiation of functions of the form \( y = f(x) \). Not all the functions will fall into this simple form. The process that we are going to cover is called implicit differentiation.
The following examples require the use of implicit differentiation. The essential skill of implicit differentiation is exactly a special case of the chain rule for derivatives. Let’s take a look at them now!

Practice Questions

Question 1

Find \( \dfrac{dy}{dx} \) for \( xy = 1 \).

Question 2

Find \( \dfrac{dy}{dx} \) for \( x^2 + y^2 = 1 \).

Question 3

Find \( \dfrac{dy}{dx} \) for \( x^4y^5 = y + 1 \).

Question 4

Find \( \dfrac{dy}{dx} \) for \( \sin{y} = x \).

Question 5

Differentiate \( \cos{xy} \) in terms of \( x \).

Question 6

Find \( \dfrac{dy}{dx} \) for \( \cos^2{x} + \cos^2{y} = \cos{(x+y)} \).


This implicit differentiation skill is useful for solving differential equations.