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The Chain Rule

( \newcommand{\kernel}{\mathrm{null}\,}\)

Our goal is to differentiate functions such as

y = (3x + 1)^{10} \nonumber

The last thing that we would want to do is FOIL this out ten times. We now look for a better way.

Definition: The Chain Rule

If y = y(u) is a function of u, and u = u(x) is a function of x then \dfrac{dy}{dx} = \dfrac{dy}{du} \dfrac{du}{dx} \nonumber

In our example we have

y = u^{10} \nonumber

and

u = 3x + 1 \nonumber

so that

\dfrac{dy}{dx} = \dfrac{dy}{du} \dfrac{du}{dx} \nonumber

= (10u9)(3) = 30(3x+1)9

Proof: Chain Rule

Recall an alternate definition of the derivative:

chain.1.gif

Example 2

Find f '(x) if

f(x) = (x^4 - 3x^3 + x)^5 \nonumber

Solution

Here

f(u) = u^5 \nonumber

and

u(x) = x^4 - 3x^3 + x \nonumber

So that the derivative is

(5u4)(4x3 - 9x2 + 1) = [5(x4 - 3x3 + x)4](4x3 - 9x2 + 1)

Example 3

Find f '(x) if

f(x) = (x^3 - x + 1)^{20} \nonumber

Solution

Here

f(u) = u^{20} \nonumber

and

u(x) = x^3 - x + 1 \nonumber

So that the derivative is

(20u^{19})(3x^2 - 1) = \left[20(x^3 - x + 1)^{19}\right](3x^2 - 1) \nonumber

Example 4

Find f '(x) if

f(x) = (1 - x)^9 (1-x^2)^4 \nonumber

Solution

Here we need both the product and the chain rule. First the product rule

f '(x) = [(1 - x)9][(1 - x2)4] ' + [(1 - x)9]' [(1 - x2)4]

Now compute

[(1 - x2)4]' = [4(1 - x2)3](-2x)

and

[(1 - x)9]' = [9(1 - x)8](-1)

Putting this all together gives

f '(x) = [(1 - x)9][4(1 - x2)3](-2x) - [9(1 - x)8] [(1 - x2)4]

Example 5

Find f '(x) if

f(x)= \dfrac{ (x^3 + 4x - 3)^7}{ (2x - 1)^3} \nonumber

Solution

Here we need both the quotient and the chain rule.

f'(x) = \dfrac{(2x - 1)^3\left[(x^3 + 4x - 3)^7\right]' - (x^3 + 4x - 3)^7 \left[(2x - 1)^3\right]'}{(2x - 1)^6} \nonumber

We first compute

[(x3 + 4x - 3)7]' = [7(x3 + 4x - 3)6](3x2 + 4)

and

[(2x - 1)3]' = [3(2x - 1)2](2)

Putting this all together gives

f](x) = \dfrac{7(2x - 1)^3(x^3 + 4x - 3)^6(3x^2 + 4) + 6(x^3 + 4x - 3)^7 (2x - 1)^2}{ (2x - 1)^6 } \nonumber

f(x)= \dfrac{x^2(5 - x^3)^4 }{3 - x} \nonumber

 


This page titled The Chain Rule is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.

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