attempt 1: 10 attempts remaining. compute the derivative of the function $y = \\frac{50}{(x^{2}+2x +…

Explanation:

Step1: Rewrite the function

Rewrite $y = \frac{50}{(x^{2}+2x + 5)^{6}}$ as $y=50(x^{2}+2x + 5)^{-6}$. Let $u=x^{2}+2x + 5$, then $y = 50u^{-6}$.

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

Using the power - rule for differentiation $\frac{d}{du}(au^{n})=nau^{n - 1}$, where $a = 50$ and $n=-6$. So, $\frac{dy}{du}=50\times(-6)u^{-7}=-300u^{-7}$.

Step3: Find $\frac{du}{dx}$

Differentiate $u=x^{2}+2x + 5$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we get $\frac{du}{dx}=2x + 2$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=-300u^{-7}$ and $\frac{du}{dx}=2x + 2$ into the chain - rule formula. Then substitute $u=x^{2}+2x + 5$ back in. $\frac{dy}{dx}=-300(x^{2}+2x + 5)^{-7}\cdot(2x + 2)=-\frac{300(2x + 2)}{(x^{2}+2x + 5)^{7}}=-\frac{600(x + 1)}{(x^{2}+2x + 5)^{7}}$.

Answer:

$-\frac{600(x + 1)}{(x^{2}+2x + 5)^{7}}$