find f(x).\nf(x)=(x^6 + 5)^{-2}\nf(x)=□

Explanation:

Step1: Apply chain - rule

Let $u = x^{6}+5$, then $y = u^{-2}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$. $\frac{dy}{du}=-2u^{-3}$

Step2: Find $\frac{du}{dx}$

Since $u = x^{6}+5$, then $\frac{du}{dx}=6x^{5}$

Step3: Calculate $\frac{dy}{dx}$

Substitute $u = x^{6}+5$ back into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$. $\frac{dy}{dx}=-2(x^{6}+5)^{-3}\cdot6x^{5}$

Step4: Simplify the expression

$f^{\prime}(x)=-12x^{5}(x^{6}+5)^{-3}=-\frac{12x^{5}}{(x^{6}+5)^{3}}$

Answer:

$-\frac{12x^{5}}{(x^{6}+5)^{3}}$