Question

find the derivative of $sqrt{1 - x^{3}}$

Explanation:

Step1: Rewrite the function

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

Step2: Apply the chain - rule

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. Let $u = 1 - x^{3}$, so $y = u^{\frac{1}{2}}$. First, find the derivative of $y$ with respect to $u$: $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}$. Then find the derivative of $u$ with respect to $x$: $\frac{du}{dx}=-3x^{2}$.

Step3: Calculate the derivative of $y$ with respect to $x$

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$ into the chain - rule formula: $\frac{dy}{dx}=\frac{1}{2}(1 - x^{3})^{-\frac{1}{2}}\cdot(-3x^{2})$.

Step4: Simplify the result

$\frac{dy}{dx}=-\frac{3x^{2}}{2\sqrt{1 - x^{3}}}$.

Answer:

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