The
chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. For example, if a composite function
f(
x) is defined as
Note that because two functions,
g and
h, make up the composite function
f, you have to consider the derivatives
g′ and
h′ in differentiating
f(
x).
If a composite function
r(
x) is defined as
Here, three functions—
m,
n, and
p—make up the composition function
r; hence, you have to consider the derivatives
m′,
n′, and
p′ in differentiating
r(
x). A technique that is sometimes suggested for differentiating composite functions is to work from the “outside to the inside” functions to establish a sequence for each of the derivatives that must be taken.
Example 1: Find
f′(
x) if
f(
x) = (3x2 + 5x − 2)8.
Example 2: Find
f′(
x) if
f(
x) = tan (sec
x).
Example 3: Find
if
y = sin3 (3
x − 1).
Example 4: Find
f′(2) if
.
Example 5: Find the slope of the tangent line to a curve
y = (
x2 − 3)5 at the point (−1, −32).
Because the slope of the tangent line to a curve is the derivative, you find that
which represents the slope of the tangent line at the point (−1,−32).
Review Topics
The Derivative
