use the rules of derivatives to calculate the derivative of the following function and simplify if possible…

Explanation:

Step1: Rewrite the function

Rewrite $\sqrt[3]{x^{4}}$ as $x^{\frac{4}{3}}$, so $g(x)=9x + x^{\frac{4}{3}}$.

Step2: Apply sum - rule of derivatives

The derivative of a sum of functions $(u + v)'=u'+v'$. Here $u = 9x$ and $v=x^{\frac{4}{3}}$.

Step3: Differentiate $u = 9x$

Using the power - rule $(ax)'=a$ for $a = 9$, we get $u'=(9x)'=9$.

Step4: Differentiate $v=x^{\frac{4}{3}}$

Using the power - rule $(x^{n})'=nx^{n - 1}$, for $n=\frac{4}{3}$, we have $v'=\frac{4}{3}x^{\frac{4}{3}-1}=\frac{4}{3}x^{\frac{1}{3}}$.

Step5: Find $g'(x)$

$g'(x)=u'+v'=9+\frac{4}{3}x^{\frac{1}{3}}$.

Answer:

$9+\frac{4}{3}x^{\frac{1}{3}}$