MATH 138 Lecture #10 - Section 3.4

1. The 3.4 Chain Rule

2. 2
Composite Functions

3. 3
The Chain Rule

4. 4
Example 1 – Using the Chain Rule
Find F '(x) if F (x) = .
Solution 1:
F (x) = (f ° g)(x) = f (g(x)) where f (u) = and g(x) = x2 + 1.
Since
and g¢(x) = 2x
we have F ¢(x) = f ¢(g(x))  g¢(x)

5. 5
Chain Rule - Exercise

6. Generalizing Differentiation Rules
In general, if y = sin u, where u is a differentiable function of
x, then, by the Chain Rule,
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Thus
In a similar fashion, all of the formulas for differentiating
functions can be combined with the Chain Rule.

7. 7
Generalized Power Rule

8. Example 3 – Using the Chain Rule with the Power Rule
Differentiate y = (x3 – 1)100.
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Solution:
Taking u = g(x) = x3 – 1 and n = 100 in (4), we have
= (x3 – 1)100
= 100(x3 – 1)99 (x3 – 1)
= 100(x3 – 1)99  3x2
= 300x2(x3 – 1)99

9. 9
Power Rule - Exercise

10. 10
Exponentials base other than e
We can use the Chain Rule to differentiate an exponential
function with any base a > 0. Recall that a = eln a. So
ax = (eln a)x = e(ln a)x
and the Chain Rule gives
(ax) = (e(ln a)x) = e(ln a)x (ln a)x
= e(ln a)x  ln a = ax ln a
because ln a is a constant. So we have the formula

11. 11
Exponentials - Example

12. The Chain Rule-Multiple Iterations
Suppose that y = f(u), u = g(x), and x = h(t), where f, g, and h
are differentiable functions.
Then, to compute the derivative of y with respect to t, we use
the Chain Rule twice:
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13. 13
Multiple Chain Rule - Example