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 \( \dfrac{dy}{dx} \). As an example we know how to find \( \dfrac{dy}{dx} \) if \( y = 2 x^3 - 2 x + 1 \).
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 \( \dfrac{dy}{dx} \). For example, we are given an equation that relates \( y \) and \( x \) as follows: \( x y + y^2 = 1 \) and asked to find \( \dfrac{dy}{dx} \).
The main idea of implicit differentiation is to differentiate both sides of the given equation and then solve the new equation obtained to find \( \dfrac{dy}{dx} \).

Implicit Differentiation Examples

Examples

Example 1

Use implicit differentiation to find the derivative \( \dfrac{dy}{dx} \) where \( y x + \sin y = 1 \)
Solution to Example 1:

Example 2

Use implicit differentiation to find the derivative \( \dfrac{dy}{dx} \) 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:

Exercises

Use implicit differentiation to find \( \dfrac{dy}{dx} \) 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) \( \dfrac{dy}{dx} = - \dfrac{1}{x} \)
2) \( \dfrac{dy}{dx} = - \dfrac{x}{y} \)
3) \( \dfrac{dy}{dx} = \dfrac{1 - \sin(x y) -x y \cos(x y)}{x^{2} \cos(x y)} \)

More References and links

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