The Chain Rule3.4

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Composite Functions

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The Chain Rule

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Example 1 – Using the Chain RuleFind 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)

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Chain Rule - Exercise

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Generalizing Differentiation RulesIn general, if y = sin u, where u is a differentiable function of x, then, by the Chain Rule,

Thus

In a similar fashion, all of the formulas for differentiating functions can be combined with the Chain Rule.

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Generalized Power Rule

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Example 3 – Using the Chain Rule with the Power Rule

Differentiate y = (x3 – 1)100.

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

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Power Rule - Exercise

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Exponentials base other than eWe 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

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Exponentials - Example

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The Chain Rule-Multiple IterationsSuppose 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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Multiple Chain Rule - Example