Question

find f(x) and simplify.
f(x)=\frac{x}{x + 50}
which of the following shows the correct application of the quotient rule?
a. \frac{(x + 50)(1)-(x)(1)}{x + 50^2}
b. \frac{(x)(1)-(x + 50)(1)}{x^2}
c. \frac{(x)(1)-(x + 50)(1)}{x + 50^2}
d. \frac{(x + 50)(1)-(x)(1)}{x^2}

Explanation:

Step1: Recall quotient - rule

The quotient - rule for a function $y=\frac{u(x)}{v(x)}$ is $y^\prime=\frac{v(x)u^\prime(x)-u(x)v^\prime(x)}{[v(x)]^2}$. For $f(x)=\frac{x}{x + 50}$, let $u(x)=x$ and $v(x)=x + 50$.

Step2: Find $u^\prime(x)$ and $v^\prime(x)$

If $u(x)=x$, then $u^\prime(x)=1$. If $v(x)=x + 50$, then $v^\prime(x)=1$.

Step3: Apply quotient - rule

Substitute $u(x),v(x),u^\prime(x),v^\prime(x)$ into the quotient - rule formula: $f^\prime(x)=\frac{(x + 50)(1)-(x)(1)}{(x + 50)^2}$.

Answer:

A. $\frac{(x + 50)(1)-(x)(1)}{(x + 50)^2}$