Question

question consider the function f(x)=x^{x/3}. find f(x), then evaluate f(1). select the correct answer below: f(1)=2 f(1)=1 f(1)=\frac{1}{3} f(1)=\frac{1}{6}

Explanation:

Step1: Use logarithmic - differentiation

Let $y = x^{x/3}$, then $\ln y=\frac{x}{3}\ln x$.
Differentiate both sides: $\frac{y'}{y}=\frac{1}{3}\ln x+\frac{x}{3}\cdot\frac{1}{x}=\frac{1}{3}\ln x+\frac{1}{3}$.
So $y' = f'(x)=x^{x/3}(\frac{1}{3}\ln x+\frac{1}{3})$.

Step2: Evaluate at $x = 1$

Substitute $x = 1$ into $f'(x)$. Since $\ln1 = 0$, then $f'(1)=1^{1/3}(\frac{1}{3}\cdot0+\frac{1}{3})=\frac{1}{3}$.

Answer:

$f'(1)=\frac{1}{3}$