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Mathematics LibreTexts

The Chain Rule

Our goal is to differentiate functions such as

\[  y = (3x + 1)^{10}  \]

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 \[\dfrac{dy}{dx} = \dfrac{dy}{du} \dfrac{du}{dx} \]

In our example we have

\[         y = u^{10} \]

and

\[  u = 3x + 1 \]

so that

\[\dfrac{dy}{dx} = \dfrac{dy}{du} \dfrac{du}{dx} \]

        = (10u9)(3) = 30(3x+1)9

 

Proof: Chain Rule

Recall an alternate definition of the derivative:

Example 2

Find f '(x) if

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

Solution

Here

\[  f(u) = u^20 \]

and

\[ u(x) = x^3 - x + 1 \]

So that the derivative is

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

Example 3

Find f '(x) if

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

Solution

Here

\[ f(u) = u^5 \]

and

\[ u(x) = x^4 - 3x^3 + x \]

So that the derivative is

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

Example 4

Find f '(x) if

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

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

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

Solution

Here we need both the quotient and the chain rule.

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

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

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

Exercise

Find the derivative of

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

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