Question

1. (2 points) find the derivative of each function. pick 1.

a. $f(x)=sqrt{1 - x^{3}}$
b. $g(x)=3^{5x - 2}$

Explanation:

Step1: Rewrite the function

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

Step2: Apply the chain - rule

The chain - rule states that if $y = u^{\frac{1}{2}}$ and $u = 1 - x^{3}$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$:
If $y = u^{\frac{1}{2}}$, then $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}$. Second, find $\frac{du}{dx}$:
If $u = 1 - x^{3}$, then $\frac{du}{dx}=-3x^{2}$.

Step3: Calculate the derivative

By the chain - rule $\frac{dy}{dx}=\frac{1}{2}(1 - x^{3})^{-\frac{1}{2}}\cdot(-3x^{2})=-\frac{3x^{2}}{2\sqrt{1 - x^{3}}}$.

Answer:

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