overall accuracy: 83.3% record: 12 score: 12 d/dx (1/(5 - 2x)) equals 2/(5 - 2x)^2 -2/(5 - 2x)^2 -2 ln|5…

Explanation:

Step1: Rewrite the function

Rewrite $\frac{1}{5 - 2x}$ as $(5 - 2x)^{-1}$.

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 $u = 5 - 2x$, so $y = u^{-1}$. First, find $\frac{dy}{du}$ and $\frac{du}{dx}$. $\frac{dy}{du}=-u^{-2}=-\frac{1}{u^{2}}$ and $\frac{du}{dx}=-2$.

Step3: Calculate the derivative

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 5 - 2x$ back in: $\frac{dy}{dx}=-\frac{1}{(5 - 2x)^{2}}\cdot(-2)=\frac{2}{(5 - 2x)^{2}}$.

Answer:

$\frac{2}{(5 - 2x)^{2}}$