Derivatives of Functions

Implicit Differentiation

Find the derivative of y with respect to x for an equation where y is defined implicitly.

Example

Differentiate both sides and isolate dy over dx.

highlighted = computed this step

Step 1 — Set up

Start with the implicit equation equal to 25.

x2+y2=25x^{2}+y^{2}=25

Step 2 — Differentiate both sides

Differentiate both sides with respect to x.

2x+2y⋅y′=0\hlmath{2x+2y\cdot y'=0}

Step 3 — Isolate the y-prime term

Move the x term to isolate the y-prime term.

2y⋅y′=−2x\hlmath{2y\cdot y'=-2x}

Step 4 — Divide by 2y

Divide by 2y to solve for y prime.

y′=−xyy'= \hlmath{-\frac{x}{y}}
implicit-differentiation Implicit differentiation is used when it is difficult to solve an equation for y in terms of x. Differentiate both sides of the equation with respect to x, treating y as a function of x, and then solve for y'.