Problem 54 - 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. $$ f(x)=\frac{3}{\left(x^{3}-4\right)^{2}} $$

Step-by-Step Solution

Answer

The derivative of the function \( f(x) = 3/(x^3 - 4)^2\) is \( f'(x) = -18x^2 (x^3 - 4)^{-3}\).

Step 1 - Differentiate treating \( f(x) = 3g(x)^{-2} \) using the power rule

Use the power rule to differentiate \( f(x) = 3g(x)^{-2} \), which gives \( f'(x) = -2 * 3g(x)^{-3} * g'(x) \). Now we need to differentiate \( g(x) = x^3 - 4 \) to find \( g'(x) \).

Step 2 - Differentiate \( g(x) = x^3 - 4 \)

The derivative of \( g(x) = x^3 - 4 \) is \( g'(x)=3x^2 \), using the power rule of differentiation. Substitute \( g'(x) \) back into the result from Step 1.

Step 3 - Substitute \( g'(x) \) into the derivative formula

Substitute \( g'(x) = 3x^2 \) into the result from Step 1, yielding \( f'(x) = -6 * 3x^2 * (x^3 - 4)^{-3} \).

Step 4 - Simplification

Simplify the expression to obtain the final answer: \( f'(x) = -18x^2 (x^3 - 4)^{-3}\)