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Chapter 16

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Chapter 16

Section 16.3

The Mean-Value Theorem; The Chain Rule

Chain Rule for a 1 Variable Function

The derivative of a complicated function that can be formed by substitutions (compositions) of simpler functions can be found by apply the chain rule. That is taking the derivatives of each simpler function and multiplying them together.

Dependence tree

Example

The variables y, u, v, and x are related by the equations below. Find .

Formula with derivatives:

y

y depends on u

u

u depends on v

Formula with Variables:

v

v depends on x

x

Chain Rule for a 2 Variable Function

If and and then ultimately the variable z is a function of the variables s and t. The variable z has 2 partial derivatives one with respect to s and the other t.

Dependence tree

z

x

y

s

t

t

s

Partial Derivative Formula (respect to s):

Partial Derivative Formula (respect to t):

Example

The variables u, v, w, x, y, and z all depend on each other as given to the right. Draw the dependence tree and give the partial derivatives using partial derivative and using the variables.

Partial Derivative Formula:

Partial Derivative Formula:

Dependence tree

w

With the variables:

With the variables:

z

x

y

v

u

v

u

v

Notice each position in the tree that ends with the independent variable that you are taking the derivative with respect to represents a term in the partial derivative. Each tier (level) of the tree represents a factor in that term.

Example

Find the derivative of the function along the curve that are given to the right.

Dependence tree

Derivative Formula using partial derivatives:

z

Derivative Formula using the variables:

x

y

t

t

Example

The variables u, v, w, x, y, z, r, and all depend on each other as given to the right. Find formulas for and using both partial derivatives and the variables.

Dependence tree

w

Partial derivative formula for :

z

y

x

u

r

u

v

Partial derivative with the variables:

r

Partial derivative formula for :

Partial derivative with variables: