Implicit Differentiation

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'.

Example:
If x² + * y*² = 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:

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

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

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Implicit Differentiation

Examples:

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

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Examples of Implicit Differentiation

1. x³ + y³ = xy
2. (x²y) + (xy²) = 3x

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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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