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 $\frac{ds}{dt}=\frac{u'v - uv'}{v^{2}}$, where $u = t$, $u'=1$, $v = 5t + 6$, and $v'=5$.

Step2: Substitute values into the formula

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

Step3: Simplify 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:

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