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Chapter 7: Problem 54

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{3}{\left(x^{3}-4\right)^{2}} $$

Short Answer

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

Step by step solution

01

- 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 \( g'(x) \).
02

- 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.
03

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

Substitute \( g'(x) = 3x^2 \) into the result from Step 1, yielding \( f'(x) = -6 * 3x^2 * (x^3 - 4)^{-3} \).
04

- Simplification

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

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Most popular questions from this chapter

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\sqrt[3]{8^{2}} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\left(\frac{6-5 x}{x^{2}-1}\right)^{2} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\left(x^{2}-2 x+3\right)^{3} $$

Use the General Power Rule to find the derivative of the function. $$ y=2 \sqrt{4-x^{2}} $$

Use the General Power Rule to find the derivative of the function. $$ s(t)=\sqrt{2 t^{2}+5 t+2} $$
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