# 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:

y^{4}+ 2x^{2}y^{2 }+ 6x^{2}= 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 *y*^{4} + *x*^{5} −
7*x*^{2} −
5*x*^{-1 }= 0.

### Example 2

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

2

y+ 5 −x^{2}−y^{3 }= 0.

### Example 3 (Involves Product Rule)

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

y^{4}+ 2x^{2}y^{2 }+ 6x^{2}= 7

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

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