Implicit Differentiation
In this lesson, we will learn how 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:
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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