WEEK 11-CHAIN RULE AND IMPLICIT FUNCTIONS

Chain Rule and Implicit Functions

Objectives

Start

Story

The Chain Rule

Implicit Differentiation

Practice

Summary

Story

The Chain Rule

Before moving on, let's review what implicit functions are in practice.

Composite functions are functions inside others.

Example: is a composite function.

y = f(g(x))

Where:

External:

Internal:

Chain Rule Differentiates Composite Functions

If , then the derivative of with respect to is:

y = f(g(x))

Example: Differentiate

y = (3x² + 2)⁵

f'(g(x))

Here:

g'(x)

Implicit Differentiation

Have you found equations where it is difficult to solve y in terms of x?

Equation where y is not solved for x are called implicit function, in which .

y=y(x)

Example 1: is an implicit function.

Example 2: is also an implicit function.

x²+y² = 9

5x²+6xy+5y² = 2

How to derivate functions where y cannot be easy solved?

A good method is to differentiate both side of the equation and solve for .

y'

Example: Find y' for the implicit function:

x² + y² = 25

Practice

Try it by yourself:

Try it by yourself:

Try it by yourself:

Try it by yourself:

Summary: Chain rule and Implicit Derivatives

implicit differentiation

chain rule

Great job!

See you next time

Welcome 6th graders!

A journey soon begin through Social Science experiences!

6TH-INTRODUCTIONTORATIONALNUMBERS-EN © 2024 by CASURID is licensed under CC BY-NC-ND 4.0

MA.912.C.2 Develop an understanding for and determine derivatives. MA.912.C.2.5 Find the derivatives of implicitly defined functions. MA.912.C.2.6 Find derivatives of inverse functions. MA.912.C.2.8 Find derivatives using logarithmic differentiation. ELA.K12.EE.1: Cite evidence to explain and justify reasoning. ELA.K12.EE.1.1 Cite evidence to explain and justify reasoning.

MATERIAL

It is highly advised to have:

• Grid paper.
• Pencils of different colors.
• Eraser.
• A rule.
• A calculator.
• Geogebra installed on your phone/tablet/computer (or use online version).