Question

if (f(w)=sqrt{5 - 4w^{3}}), find (f(w)). use exact values.
(f(w)=)

Explanation:

Step1: Rewrite the function

Rewrite $f(w)=\sqrt{5 - 4w^{3}}$ as $f(w)=(5 - 4w^{3})^{\frac{1}{2}}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = u^{\frac{1}{2}}$ and $u = 5-4w^{3}$, then $\frac{dy}{dw}=\frac{dy}{du}\cdot\frac{du}{dw}$. First, find $\frac{dy}{du}$: $\frac{d}{du}(u^{\frac{1}{2}})=\frac{1}{2}u^{-\frac{1}{2}}$. Second, find $\frac{du}{dw}$: $\frac{d}{dw}(5 - 4w^{3})=-12w^{2}$.

Step3: Calculate $f^{\prime}(w)$

Substitute $u = 5 - 4w^{3}$ back into $\frac{dy}{du}$ and multiply by $\frac{du}{dw}$. So $f^{\prime}(w)=\frac{1}{2}(5 - 4w^{3})^{-\frac{1}{2}}\cdot(-12w^{2})$.
Simplify the expression: $f^{\prime}(w)=\frac{-6w^{2}}{\sqrt{5 - 4w^{3}}}$.

Answer:

$\frac{-6w^{2}}{\sqrt{5 - 4w^{3}}}$