Calculus

Section 2.5

Implicit Differentiation

Terminology

Equations in explicit form can be solved for y in terms of x (e.g. functions)

Equations in implicit form CANNOT be solved for y in terms of x (e.g. equations of circles)

Implicit Differentiation

Goal is to find (to derive with respect to x) Differentiate x terms as usual (apply Power Rule,

etc.) Differentiate y terms, applying the Chain Rule

dy

dx

ex. d y 4 dx

d y 4 dx

4y 3 dydx

Implicit Differentiation Process

1. Differentiate both sides of the equation with respect to x

2. Move all terms to the left side, and all other terms to the right side

3. Factor out from the left side

4. Solve for , by dividing

dy

dx

dy

dx

dy

dx

Example

Find dydx : y 3 y 2 5y x 2 4

3y 2 dydx 2y dydx 5 dy

dx 2x 0

dydx 3y 2 2y 5 2x

3y 2 dydx 2y dydx 5 dy

dx 2x

dydx

2x

3y 2 2y 5

Example 2

Find dydx : x 2 4y 2 y 2x 4

2x 8y dydx 2y dydx x y 2 0

2x 8y dydx 2y dydx x y 2 0

8y dydx 2y dydx x y 2 2x

dydx 8y 2xy y 2 2x

Example 2 continued

dydx 8y 2xy y 2 2x

dydx

y 2 2x

8y 2xy

Example

Find d2y

dx 2 : dydx y 2

3x

d 2y

dx 2 3x(2y dydx ) y

2(3)

(3x)2

d 2y

dx 2 6xy dydx 3y 2

(3x)2

We want the derivative in terms of x and y, so substitute for dy/dx

Ex. Cont.

d 2y

dx 2 6xy

y 2

3x

3y 2

(3x)2

d 2y

dx 2 2y 3 3y 2

(3x)2

Double Derivatives