Question

quiz 4 - requires respondus lockdown browser + webcam
started: sep 15 at 2:41pm
quiz instructions
access code: start
timed: 30 minutes
number of attempts: 1
lockdown browser and respondus monitor required.
this quiz covers material up through week 4, where section 2.3 was covered. you may still need to know material weeks.
question 4
find the derivative of (f(x)=\frac{qx + r}{sx + t}).
(\frac{qt+rs}{(sx + t)^2})
(\frac{qt - rs}{(sx - t)^2})
(\frac{rs - qt}{(sx + t)^2})
the correct answer is not listed.
(\frac{qt - rs}{(sx + t)^2})

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. For $f(x)=\frac{qx + r}{sx + t}$, let $u = qx + r$ and $v=sx + t$.

Step2: Find $u^\prime$ and $v^\prime$

Differentiate $u$ with respect to $x$: $u^\prime=\frac{d}{dx}(qx + r)=q$. Differentiate $v$ with respect to $x$: $v^\prime=\frac{d}{dx}(sx + t)=s$.

Step3: Apply quotient - rule

$f^\prime(x)=\frac{u^\prime v - uv^\prime}{v^{2}}=\frac{q(sx + t)-(qx + r)s}{(sx + t)^{2}}$.

Step4: Expand and simplify

Expand the numerator: $q(sx + t)-(qx + r)s=qsx+qt - qsx-rs=qt - rs$. So $f^\prime(x)=\frac{qt - rs}{(sx + t)^{2}}$.

Answer:

E. $\frac{qt - rs}{(sx + t)^{2}}$