question 3 (1 point)\nthe limit $lim_{h\rightarrow0}\frac{(x + h)^{1/3}-x^{1/3}}{h}$ is,\n$\frac{1}{3x^{2/3}}…

question 3 (1 point)\nthe limit $lim_{h\rightarrow0}\frac{(x + h)^{1/3}-x^{1/3}}{h}$ is,\n$\frac{1}{3x^{2/3}}$\n$\frac{2}{3}x^{1/3}$\n$x^{2/3}$\ndne\n$x^{1/3}$

question 3 (1 point)\nthe limit $lim_{h\rightarrow0}\frac{(x + h)^{1/3}-x^{1/3}}{h}$ is,\n$\frac{1}{3x^{2/3}}$\n$\frac{2}{3}x^{1/3}$\n$x^{2/3}$\ndne\n$x^{1/3}$

Answer

Explanation:

Step1: Recall the definition of the derivative

The given limit $\lim_{h\rightarrow0}\frac{(x + h)^{1/3}-x^{1/3}}{h}$ is in the form of the definition of the derivative $f^\prime(x)=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}$, where $f(x)=x^{1/3}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For $y=x^{1/3}$, using the power - rule $n=\frac{1}{3}$, we have $y^\prime=\frac{1}{3}x^{\frac{1}{3}-1}$.

Step3: Simplify the exponent

$\frac{1}{3}x^{\frac{1}{3}-1}=\frac{1}{3}x^{\frac{1 - 3}{3}}=\frac{1}{3}x^{-2/3}=\frac{1}{3x^{2/3}}$.

Answer:

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