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Mathematical Expression Editor
We compute the derivative of a composition.
1 The Chain Rule
Compositions are common in typical looking functions. For example, the hypotensue
of a right triangle with sides and is . This function is a composition of the inside
polynomial, and the square root on the outside, . Consider an exponential
growth model with parameter , , where . The expression is the composition
of the inner linear function and the outer, exponential function . In this
section, we will learn how to differentiate such functions.
Chain Rule If
and are differentiable functions and if , then This can also be written as
or in Leibniz notation as where and .
In words, the derivative of a composition is the derivative of the outside (with the
inside left in), times the derivative of the inside.
example 1 Find if . We write as a composition: with
To find we need and . We have, By the chain rule, We can also solve this problem
using Leibniz notation. In that case, we let , and . Substituting for in , we can write
Then the derivatives are Finally, the chain rule says to multiply thses derivatives:
Here is a video of Example 1
_
(problem 1)
Compute
The chain rule says:
The “outside” function is and the “inside” function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 2
Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 2)
Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 3 Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 3) Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 4
Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 4) Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 5
Find if . We write as a composition: with To find we need and . We have, By the chain
rule, \begin{align*} h'(x) &= f'(g(x))g'(x)\\ &= \frac{1}{2\sqrt{x^2 + 4}}\cdot 2x\\ &= \frac{2x}{2\sqrt{x^2 + 4}}\\ =\frac{x}{\sqrt{x^2 + 4}}. \end{align*}
Here is a video of the example
_
(problem 5) Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 6
Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 6a) Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
(problem 6b) What is the derivative of ?
example 7 Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 7) Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 8 Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 8) Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 9 Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 9) Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 10 Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 10)
Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 11 Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 11) Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 12 Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 12)
Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 13 Find if . We write as a composition: with To find we need and . We have, By the chain
rule,
Here is a video of the example
_
(problem 13a) Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
(problem 13b) Compute
The chain rule says:
The outside function is and the inside function is .
Leave the inside in,
Multiply by the derivative of the inside,
example 14 Find . We use the chain rule with as the inside function and we get
The above formula is both useful and important. We will see it again in the
Logarithmic Differentiation section.
Find .
We use the above formula with to get
(problem 14) Compute
Here are some detailed, lecture style videos on the chain rule: