find the derivative of f(x). f(x) = √(x³) write your answer as a constant times a power of x. f(x) =

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\sqrt{x^{3}}$ as $f(x)=x^{\frac{3}{2}}$ using the rule $\sqrt[n]{a^{m}}=a^{\frac{m}{n}}$.

Step2: Apply the power - rule

The power - rule for differentiation is $\frac{d}{dx}(x^{n})=nx^{n - 1}$. For $n=\frac{3}{2}$, we have $f^\prime(x)=\frac{3}{2}x^{\frac{3}{2}-1}$.

Step3: Simplify the exponent

$\frac{3}{2}-1=\frac{3 - 2}{2}=\frac{1}{2}$. So $f^\prime(x)=\frac{3}{2}x^{\frac{1}{2}}$.

Answer:

$\frac{3}{2}x^{\frac{1}{2}}$