Question

hw13 derivatives of trigonometric
score: 6/7 answered: 6/7
question 7
find $\frac{dy}{dv}$ for the given function.
$y = \frac{2 - cos(v)}{7 + 4cos(v)}$
$\frac{dy}{dv}=$
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Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $\frac{dy}{dv}=\frac{u'v - uv'}{v^{2}}$. Here, $u = 2-\cos(v)$ and $v=7 + 4\cos(v)$. First, find $u'$ and $v'$.
$u'=\frac{d}{dv}(2-\cos(v))=\sin(v)$ (since $\frac{d}{dv}(2)=0$ and $\frac{d}{dv}(-\cos(v))=\sin(v)$) and $v'=\frac{d}{dv}(7 + 4\cos(v))=-4\sin(v)$ (since $\frac{d}{dv}(7)=0$ and $\frac{d}{dv}(4\cos(v))=-4\sin(v)$).

Step2: Substitute into quotient - rule formula

$\frac{dy}{dv}=\frac{\sin(v)(7 + 4\cos(v))-(2-\cos(v))(-4\sin(v))}{(7 + 4\cos(v))^{2}}$.
Expand the numerator:

$$
LATEXBLOCK0
$$

So, $\frac{dy}{dv}=\frac{15\sin(v)}{(7 + 4\cos(v))^{2}}$.

Answer:

$\frac{15\sin(v)}{(7 + 4\cos(v))^{2}}$