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Find {eq}f'(x) {/eq} for

(a) {eq}\displaystyle f (x) = \frac {(x + 3)^4} {(x^2 + 2 x)} {/eq}

(b) {eq}\displaystyle f (x) = \bigg(\frac {(3 x + 1)} {x^2}\bigg)^3 {/eq}.

Question:

Find {eq}f'(x) {/eq} for

(a) {eq}\displaystyle f (x) = \frac {(x + 3)^4} {(x^2 + 2 x)} {/eq}

(b) {eq}\displaystyle f (x) = \bigg(\frac {(3 x + 1)} {x^2}\bigg)^3 {/eq}.

Rules of Differentiation:

Derivative of {eq}\displaystyle f(x) {/eq} with respect to {eq}\displaystyle x {/eq} is given by {eq}\displaystyle f'(x) {/eq}.

Formulas Used:

1.Quotient Rule of Differentiation:

{eq}\displaystyle \left( \frac{u}{v} \right)'=\frac{vu'-uv'}{v^2} {/eq}.

2.Product Rule of Differentiation:

{eq}\displaystyle (uv)'=uv'+vu' {/eq}.

3.Chain Rule of Differentiation:

{eq}\displaystyle (f(g(x)))'=f'(g(x))g'(x) {/eq}.

4.{eq}\displaystyle (x^n)'=nx^{n-1} {/eq}.

Answer and Explanation: 1

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a.

Given {eq}\displaystyle f (x) = {(x + 3)^4} / {(x^2 + 2 x)} {/eq}.

Differentiating with respect to {eq}\displaystyle x {/eq}.

Using ...

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Using the Chain Rule to Differentiate Complex Functions

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Chapter 8 / Lesson 6
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Learn how to differentiate a function using the chain rule of differentiation. Find various chain rule derivative examples with various function types.


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