Calculus · Quick Reference

Implicit Differentiation Cheat Sheet

Differential Calculus · Lesson 6/20

In one line: some equations mix x and y together so that y cannot easily be isolated. Implicit differentiation lets us find dy/dx anyway, treating y as a function of x throughout.

Key Ideas

1Explicit vs. Implicit. Explicit: y is isolated on one side (y = x^2). Implicit: x and y are mixed (x^2 + y^2 = 25).

2Differentiate Both Sides. Take d/dx of every term on both sides of the equation, remembering that y is a function of x.

3The Key Trick. Whenever you differentiate a term with y, use the chain rule and multiply by dy/dx, since y itself depends on x.

4Solving for dy/dx. After differentiating, collect all dy/dx terms on one side and solve algebraically.

5Where It's Used. Useful for curves like circles, ellipses, and other relations that are not functions in the simple y=f(x) sense.

Worked Examples

Find dy/dx for x^2 + y^2 = 25.

dy/dx = -x/y

Find dy/dx for x*y = 10.

dy/dx = -y/x

Find dy/dx for y^3 + x^2 = 4y.

dy/dx = -2x / (3y^2 - 4)

Formulas

d/dx[y^n] = n*y^(n-1) * dy/dx

Practice Yourself

Find dy/dx for x^2 + y^2 = 9.

-x/y

Find dy/dx for x^2*y = 5.

dy/dx = -2y/x

Find dy/dx for y^2 = x.

dy/dx = 1/(2y)