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}
f(x) =

Explanation:

Step1: Recall quotient - rule

The quotient rule for a function $y = \frac{u}{v}$ is $y^\prime=\frac{v u^\prime - u v^\prime}{v^{2}}$, where $u$ and $v$ are functions of $x$, and $u^\prime$ and $v^\prime$ are their derivatives with respect to $x$. For $f(x)=\frac{x}{x + 50}$, we have $u=x$, $u^\prime = 1$, $v=x + 50$, and $v^\prime=1$.

Step2: Apply quotient - rule

Substitute $u$, $u^\prime$, $v$, and $v^\prime$ into the quotient - rule formula: $f^\prime(x)=\frac{(x + 50)\times1-(x)\times1}{(x + 50)^{2}}$.

Step3: Simplify the expression

Expand the numerator: $(x + 50)\times1-(x)\times1=x + 50-x=50$. So $f^\prime(x)=\frac{50}{(x + 50)^{2}}$.

Answer:

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