Chapter 16

Section 16.3The Mean-Value Theorem; The Chain

Rule

Chain Rule for a 1 Variable FunctionThe 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.

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

Dependence treey

u

v

x

y depends on u

u depends on v

v depends on x

𝑑𝑦𝑑𝑢

𝑑𝑢𝑑𝑣𝑑𝑣𝑑𝑥

Formula with derivatives:

Formula with Variables:

Chain Rule for a 2 Variable FunctionIf 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 treez

x y

s st t

𝜕𝑧𝜕𝑥

𝜕 𝑧𝜕 𝑦

𝜕𝑥𝜕 𝑠

𝜕𝑥𝜕𝑡

𝜕 𝑦𝜕𝑡

𝜕 𝑦𝜕𝑠

Partial Derivative Formula (respect to s): Partial Derivative Formula (respect to t):

ExampleThe 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.

Dependence treew

x y z

u v u v v

Partial Derivative Formula: Partial Derivative Formula:

𝜕𝑤𝜕𝑢

=𝑦 3 ln𝑣+3 𝑥 𝑦 22𝑢With the variables: With the variables:

𝜕𝑤𝜕𝑣

=𝑦 3𝑢𝑣

+3 𝑥 𝑦 22+𝑒 𝑧 12√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.

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

Dependence treez

x y

t t

Derivative Formula using partial derivatives:

Derivative Formula using the variables:

ExampleThe 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 treew

x y z

u v r

𝜃

u

𝜃𝜃 r

Partial derivative formula for :

Partial derivative with variables:

Partial derivative formula for :

Partial derivative with the variables: