Question

differentiate ( f(x) = (8 + x^4)^{2/3} ). ( f(x) = ) input box submit answer next item

Explanation:

Step1: Identify the function type

We have \( F(x)=(8 + x^{4})^{\frac{2}{3}} \), which is a composite function. Let \( u = 8+x^{4} \), so \( F(u)=u^{\frac{2}{3}} \).

Step2: Apply the chain rule

The chain rule states that \( F^{\prime}(x)=F^{\prime}(u)\cdot u^{\prime} \). First, find the derivative of \( F(u) \) with respect to \( u \):
Using the power rule \( \frac{d}{du}(u^{n})=nu^{n - 1} \), for \( F(u)=u^{\frac{2}{3}} \), we have \( F^{\prime}(u)=\frac{2}{3}u^{\frac{2}{3}-1}=\frac{2}{3}u^{-\frac{1}{3}} \).

Step3: Find the derivative of \( u \) with respect to \( x \)

Given \( u = 8 + x^{4} \), the derivative \( u^{\prime}=\frac{d}{dx}(8+x^{4})=4x^{3} \) (using the power rule \( \frac{d}{dx}(x^{n})=nx^{n - 1} \) and the derivative of a constant is 0).

Step4: Substitute back \( u = 8 + x^{4} \) into the chain rule formula

\( F^{\prime}(x)=\frac{2}{3}(8 + x^{4})^{-\frac{1}{3}}\cdot4x^{3} \).
Simplify the expression:
\( F^{\prime}(x)=\frac{8x^{3}}{3(8 + x^{4})^{\frac{1}{3}}} \) or \( F^{\prime}(x)=\frac{8x^{3}}{3\sqrt[3]{8 + x^{4}}} \)

Answer:

\(\frac{8x^{3}}{3\sqrt[3]{8 + x^{4}}}\) (or \(\frac{8x^{3}}{3(8 + x^{4})^{\frac{1}{3}}}\))