Implicit Differentiation

Implicit Differentiation examples with detailed solutions are presented

Implicit Differentiation Explained

When we are given a function y explicitly in terms of x, we use the rules and formulas of differentions to find the derivative dy/dx. As an example we know how to find dy/dx if y = 2 x3 - 2 x + 1.
In some other situations, however, instead of a function given explicitly, we are given an equation including terms in y and x and we are asked to find dy/dx. For example, we are given an equation that relates y and x as follows: x y + y2 = 1 and asked to find dy/dx.
The main idea of implicit differentiation is to differentiate both sides of the given equation and then solve the new equation obtained to find dy/dx.

Implicit Differentiation Examples


Example 1

Use implicit differentiation to find the derivative dy / dx where y x + sin y = 1
Solution to Example 1:

Example 2

Use implicit differentiation to find the derivative dy / dx where y 4 + x y 2 + x = 3
Solution to Example 2:

Example 3

Find all points on the graph of the equation
x 2 + y 2 = 4

where the tangent lines are parallel to the line x + y = 2
Solution to Example 3:


Use implicit differentiation to find dy/dx for each equation given below.
1) x e y = 3
2) x 2 + y 2 = 20
3) x sin(x y) = x

Solutions to the Above Exercises

1) dy/dx = - 1 / x
2) dy/dx = - x / y
3) dy/dx = [ 1 - sin(x y) -x y cos(x y) ] / [ x2 cos(x y)]

More References and links

Tables of Formulas for Derivatives
Rules of Differentiation of Functions in Calculus
differentiation and derivatives