find f(x). f(x)=(x^3 + 2)^-5 f(x)=□

Explanation:

Step1: Identify the outer - inner functions

Let $u = x^{3}+2$, then $y = u^{-5}$.

Step2: Differentiate the outer function

Using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, for $y = u^{-5}$, we have $\frac{dy}{du}=-5u^{-6}$.

Step3: Differentiate the inner function

For $u = x^{3}+2$, $\frac{du}{dx}=3x^{2}$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=-5u^{-6}$ and $\frac{du}{dx}=3x^{2}$ and $u = x^{3}+2$ back in, we get $\frac{dy}{dx}=-5(x^{3}+2)^{-6}\cdot3x^{2}$.

Answer:

$-15x^{2}(x^{3}+2)^{-6}$