Question

find $\frac{ds}{dt}$ if $s = \frac{t}{5t + 6}$
$\frac{ds}{dt}=square$

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $s=\frac{u}{v}$, then $\frac{ds}{dt}=\frac{u'v - uv'}{v^{2}}$. Here, $u = t$, so $u'=1$, and $v = 5t + 6$, so $v'=5$.

Step2: Substitute values into quotient - rule formula

$\frac{ds}{dt}=\frac{1\times(5t + 6)-t\times5}{(5t + 6)^{2}}$.

Step3: Simplify the numerator

Expand the numerator: $1\times(5t + 6)-t\times5=5t + 6-5t$.
$5t+6 - 5t=6$.
So, $\frac{ds}{dt}=\frac{6}{(5t + 6)^{2}}$.

Answer:

$\frac{6}{(5t + 6)^{2}}$