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.

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.)

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