8. Differentiation of Implicit Functions

by M. Bourne

We meet many equations where y is not expressed explicitly in terms of x only, such as:

y4 + 2x2y2 + 6x2 = 7

You can see several examples of such expressions in the Polar Graphs section.

It is usually difficult, if not impossible, to solve for y so that we can then find `(dy)/(dx)`.

We need to be able to find derivatives of such expressions to find the rate of change of y as x changes. To do this, we need to know implicit differentiation.

Let's learn how this works in some examples.

Example 1

Find the expression for `(dy)/(dx)` if y4 + x5 − 7x2 − 5x-1 = 0.

Continues below

Example 2

Find the slope of the tangent at the point `(2,-1)` for the curve:

2y + 5 − x2y3 = 0.

Example 3 (Involves Product Rule)

Find the expression for `(dy)/(dx)` if:

y4 + 2x2y2 + 6x2 = 7

(This is the example given at the top of this page.)


Search IntMath, blog and Forum

Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Calculus Lessons on DVD


Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.