Question

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(x^{2}-9\right)^{2 / 3} $$

Short answer

The derivative of the function \( f(x) = (x^{2}-9)^{2/3} \) is \( f'(x) = (4x)/(3 (x^{2}-9)^{1/3}) \).

Step-by-step solution

Identify \(u\) and \(n\)

The function is \(f(x) = (x^{2}-9)^{2/3}\). Here, \(u = x^{2}-9\) and \(n= 2/3.\)

Apply the General Power Rule

Differentiate \( u \) to obtain \( u' \). The derivative of \( x^{2} \) is \( 2x \), and the derivative of \( -9 \) is \( 0 \). Therefore, \( u' = 2x \). Then, substitute \( u, n, u' \) into the General Power Rule to obtain \( f'(x) = n \cdot u^{n-1} \cdot u' = 2/3 \cdot (x^{2}-9)^{2/3 - 1} \cdot 2x \). Simplify to obtain \( f'(x) = 4/3 \cdot x \cdot (x^{2}-9)^{-1/3} \).

Final Simplifying

Simplify \( f'(x) = 4/3 \cdot x \cdot (x^{2}-9)^{-1/3} \) to a more standardized form \( f'(x) = (4x/3 (x^{2}-9)^{-1/3}) \).