Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ f(x)=-3 \sqrt[4]{2-9 x} $$

Step-by-Step Solution

Answer

The derivative of the function \( f(x)=-3 \sqrt[4]{2-9 x} \) is \( f'(x) = \frac{27}{4} * (2 - 9x)^{-3/4} \).

Step 1: Rewrite the Function in Power Form

The function is \( f(x)=-3 \sqrt[4]{2-9 x} \), which can be rewritten as \( f(x) = -3(2 - 9x)^{1/4} \). Now, the General Power Rule can be applied.

Step 2: Apply the General Power Rule

The General Power Rule states that the derivative of \(x^n\), where n is any real number, is \(n*x^{n-1}\). So, by applying the rule, the derivative of \( f(x) = -3(2 - 9x)^{1/4} \) is \( f'(x) = -3 * 1/4 * (2 - 9x)^{1/4 -1} \).

Step 3: Differentiate the Inside of the Function

In addition to applying the power rule, the Chain Rule also necessitates taking the derivative of the inside of the function. The derivative of \(2 - 9x\) is \(-9\). Hence, the derivative becomes \(f'(x) = -3 * 1/4 * -9 * (2 - 9x)^{-3/4}\).

Step 4: Simplify the Derivative

By doing the mathematical operations, \(f'(x) = \frac{27}{4} * (2 - 9x)^{-3/4}\) is obtained.