Question

find the derivative of the following function.
y = \frac{1}{3 + 4\cos x}
\frac{dy}{dx}=\square

Explanation:

Step1: Identify the quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 1$ and $v=3 + 4\cos x$.

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

Since $u = 1$, then $u^\prime=0$. Since $v = 3+4\cos x$, then $v^\prime=-4\sin x$ (using the derivative of a constant is 0 and $\frac{d}{dx}(\cos x)=-\sin x$).

Step3: Apply the quotient - rule

$y^\prime=\frac{0\times(3 + 4\cos x)-1\times(-4\sin x)}{(3 + 4\cos x)^{2}}$.

Step4: Simplify the expression

$y^\prime=\frac{4\sin x}{(3 + 4\cos x)^{2}}$.

Answer:

$\frac{4\sin x}{(3 + 4\cos x)^{2}}$