Example

Consider for example

y + x = -4.

This function can be differentiated normally by using algebra to change this equation to an explicit function:

f(x) = y = −x - 4;

such differentiation would result in a value of −1. Equally, one can use implicit differentiation;

dy/dx + dx/dx = 0 = dy/dx + 1; dy/dx = -1.

An example of an implicit function, for which implicit differentiation might be easier than attempting to use explicit differentiation, is

x⁴ + 2y² = 8.

In order to differentiate this explicitly , one would have to obtain (via algebra)

,

and then differentiate this function. This creates two derivatives, one for y > 0 and another for y < 0. Implicit differentiation avoids this.

One might find it substantially easier to implicitly differentiate the implicit function;

4x³ + 4y(dy/dx) = 0;

thus,

dy/dx = −4x³ / 4y = −x³ / y.