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Find the derivative of the function y = \frac {1}{4} (7x + 11)^3 + (1 - \frac {1}{x^3})^{-1}.

Question:

Find the derivative of the function {eq}\displaystyle y = \frac {1}{4} (7x + 11)^3 + (1 - \frac {1}{x^3})^{-1} {/eq}.

Chain Rule:

Suppose we are asked to differentiate the composite function {eq}y = f(g(x)) {/eq}.

The derivative of the composite function {eq}y = f(g(x)) {/eq} is obtained by utilizing the chain rule, and is given by:

{eq}y' = f'(g(x)) g'(x) {/eq}

The chain rule basically says that for nested function, we differentiate from the inside out and take their product, like multiplicative Russian dolls.

Use the power rule:

{eq}\displaystyle \frac {d}{dx} (x^n) = nx^{n - 1} {/eq}

Answer and Explanation:

Given:

{eq}y = \frac {1}{4} (7x + 11)^3 + \left (1 - \frac {1}{x^3}\right )^{-1} {/eq}

To find:

{eq}\displaystyle y' = \frac {dy}{dx} = ? {/eq}

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Applying the Rules of Differentiation to Calculate Derivatives

from Math 104: Calculus

Chapter 8 / Lesson 13
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