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Implicit Differentiation

Related Topics: More Lessons on Calculus


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'.


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

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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