Implicit Differentiation

Implicit differentiation can be used the find the derivatives of equations that are not functions.

Some functions can be described by expressing one variable explicitly in terms of another variable.

For example:
y = x² + 3
y = x cos x

However, some equations are defined implicitly by a relation between x and y.

For example:
x² + y² = 16
x² + y² = 4xy

We do not need to solve an equation for y in terms of x in order to find the derivative of y. Instead, we can use the method of implicit differentiation. This involves differentiating both sides of the equation with respect to x and then solving the resulting equation for y’.

Example:

If x² + y² = 16, find

Solution:

Step 1: Differentiate both sides of the equation

Step 2: Using the Chain Rule, we find that

Step 3: Substitute equation (2) into equation (1)

Step 4: Solve for

Example:

Find y’ if x³ + y³ = 6xy

Solution:

Videos

Implicit Differentiation - Basic Idea and Examples
The basic idea about using implicit differentiation

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