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Implicit Differentiation




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

Related Topics: More Lessons on Calculus

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:




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 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


 
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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