Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ g(x)=\frac{3}{\sqrt[3]{x^{3}-1}} $$

Step-by-Step Solution

Answer

The derivative of the given function is \( g'(x) = -3x^{2}(x^{3}-1)^{-4/3} \). The chain rule was used to find the derivative.

Step 1: Rewrite the function in a tractable form

Rewrite the function to highlight its components and simplify the differentiation process. The function can be expressed as \( g(x) = 3 \cdot (x^{3}-1)^{-1/3} \)

Step 2: Apply the chain rule

First differentiate the outer function and then differentiate the inner function. The chain rule is formulated as follows: If we have a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \). In our case, the outer function is \( f(u) = 3 \cdot u^{-1/3} \) and the inner function is \( g(x) = x^{3}-1 \). Differentiating the outer function we obtain \( f'(u) = -u^{-4/3} \) and differentiating the inner function we get \( g'(x) = 3x^{2} \). Applying the chain rule results in \( g'(x) = f'(g(x)) \cdot g'(x) = - (x^{3}-1)^{-4/3} \cdot 3x^{2} \).

Step 3: Simplify the function

Simplify your answer for the most concise form. Factor out what you can. It results in \( g'(x) = -3x^{2}(x^{3}-1)^{-4/3} \).