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.
However, some equations are defined implicitly by a relation between x and y.
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 + y2 = 16, find
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
Find y’ if x3 + y3 = 6xy
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
+ xy + cos(y) = 8y
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
Examples of Implicit Differentiation
y) + (xy2
) = 3x
How to use Implicit Differentiation to find a Derivative?
Find the second derivative using implicit differentiation
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