Question

differentiate $6x^{\frac{2}{3}}$.
a. $4x^{-\frac{2}{3}}$
b. $4x^{-\frac{1}{3}}$
c. $4x^{\frac{1}{3}}$
d. $6x^{\frac{1}{3}}$

Explanation:

Step1: Recall Power Rule

The power rule for differentiation is $\frac{d}{dx}(x^n) = nx^{n - 1}$, and for a constant multiple $c x^n$, the derivative is $c\cdot nx^{n - 1}$.

Step2: Apply Power Rule

Given the function $y = 6x^{\frac{2}{3}}$, here $c = 6$ and $n=\frac{2}{3}$.
Using the power rule, the derivative $y'$ is $6\times\frac{2}{3}x^{\frac{2}{3}- 1}$.
First, calculate $6\times\frac{2}{3}=4$.
Then, calculate the exponent: $\frac{2}{3}-1=\frac{2 - 3}{3}=-\frac{1}{3}$.
So the derivative is $4x^{-\frac{1}{3}}$.

Answer:

b. $4x^{-1/3}$