Calculus Lesson 3.2: Implicit Differentiation

​3.2: Implicit Differentiation

By J Barrientos

We will find the equation of a tangent line using implicit differentiation.

​Content Objective #2

We will find a derivative using implicit differentiation.

​Content Objective #1

Content Objectives

​We have learned multiple ways of calculating derivatives. However, most of the derivatives have come from function in explicit form.

​​Introduction

​Usually in the form 'y=' or 'f(x) = '

Explicit Form

​Usually mixed variables, or one variable cannot be clearly isolated.

​​Implicit Form

Think about the Derivative

When we take the derivative, we are looking at the change in y as x changes. But when there are multiple y's or a mix of x and y, how do we look at that change?

Implicit Differentiation

The process of finding the derivative of an implicit equation.

This will find what is known as the 'implicit derivative'.

Steps for Implicit Differentiation

Steps for Implicit Differentiation

Steps for Implicit Differentiation

Steps for Implicit Differentiation

Math Response

Find the derivative of $$3xy^2-4x=7y$$ .

y' =

Type answer here

Correct Answer: \frac{4-3y^2}{6xy-7}

Deg^°

Rad

​Implicit Derivatives with Exponential and Logarithms

​Implicit Derivatives with Exponential and Logarithms

Math Response

Find the derivative of $$\ln\left(3y^2\right)=7x^2$$ . y' =

Type answer here

Correct Answer: 7xy

Deg^°

Rad

To evaluate this derivative we will need BOTH x and y.

We can't really solely on x.

Knowing the derivative can also help us find the equation of the tangent line.

​What we need.

Just because the derivative of an implicit equation has x and y, doesn't mean we can't evaluate its derivative.

Introduction

Implicit Differentiation at a Point

Steps to Find Derivative at a Given Point

​1. Find the value of x and y
2. Find the implicit derivative.
3. Substitute your known values.
4. Solve for y'

This will also help us find the equation of the tangent line.

​Form a Horizontal Tangent Line

y ' = 0

Equation of the Tangent Line: y = #

​​Derivative of 0

Form a Vertical Tangent Line

y ' = undefined

Equation of the Tangent Line: x = #

​​Undefined Derivative

​Example #1

​Example #2

​Example #3

​Example #4

AP Practice