Implicit Differentiation

In these lessons, we will learn how implicit differentiation can be used the find the derivatives of equations that are not functions.

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Related Pages
Calculus: Derivatives
Calculus: Derivative Rules
Calculus Lessons

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'.

If x2 + * y*2 = 16, find

Step 1: Differentiate both sides of the equation

Implicit Differentiation

Step 2: Using the Chain Rule, we find that


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


Step 4: Solve for


Find y’ if x3 + y3 = 6xy

Implicit Differentiation Solution

Implicit Differentiation - Basic Idea and Examples
What is implicit differentiation?

The basic idea about using implicit differentiation

  1. Take derivative, adding dy/dx where needed.
  2. Get rid of parenthesis.
  3. Solve for dy/dx

Find dy/dx.
x2 + xy + cos(y) = 8y

Implicit Differentiation


  1. Find dy/dx
    1 + x = sin(xy2)
  2. Find the equation of the tangent line at (1, 1) on the curve x2 + xy + y2 = 3

Examples of Implicit Differentiation

  1. x3 + y3 = xy
  2. (x2y) + (xy2) = 3x

How to use Implicit Differentiation to find a Derivative?
Find the second derivative using implicit differentiation
Find yn for:
9x2 + y2 = 9

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