Question

differentiate ( f(x) = (2 + x^9)^{2/3} ). ( f(x) = ) next item

Explanation:

Step1: Apply the chain rule

The chain rule states that if we have a function \( F(x) = [g(x)]^n \), then \( F'(x) = n[g(x)]^{n - 1} \cdot g'(x) \). Here, \( g(x)=2 + x^{9} \) and \( n=\frac{2}{3} \).
First, find the derivative of the outer function: \( \frac{d}{dx}[g(x)]^n=\frac{2}{3}(2 + x^{9})^{\frac{2}{3}-1}=\frac{2}{3}(2 + x^{9})^{-\frac{1}{3}} \)

Step2: Find the derivative of the inner function

The inner function is \( g(x)=2 + x^{9} \), and its derivative \( g'(x)=\frac{d}{dx}(2 + x^{9}) = 9x^{8} \) (using the power rule \( \frac{d}{dx}x^k=kx^{k - 1} \) and the derivative of a constant is 0)

Step3: Multiply the two derivatives

Multiply the derivative of the outer function and the derivative of the inner function:
\( F'(x)=\frac{2}{3}(2 + x^{9})^{-\frac{1}{3}}\cdot9x^{8} \)
Simplify the coefficients: \( \frac{2}{3}\times9 = 6 \)
So, \( F'(x)=6x^{8}(2 + x^{9})^{-\frac{1}{3}} \) or we can write it as \( F'(x)=\frac{6x^{8}}{\sqrt[3]{2 + x^{9}}} \)

Answer:

\( 6x^{8}(2 + x^{9})^{-\frac{1}{3}} \) (or \( \frac{6x^{8}}{\sqrt[3]{2 + x^{9}}} \))