Question

Find the derivative of the following functions(it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $\frac{\cos x}{1+\sin x}$

Answer 1

Let $f\left(x\right)=\frac{\cos x}{1+\sin x}$
Thus by quotient rule,
$f^{\prime }\left(x\right)=\frac{(1+\sin x)\frac{d}{dx}\left(\cos x\right)-\left(\cos x\right)\frac{d}{dx}(1+\sin x)}{{(1+\sin x)}^2}$
$=\frac{(1+\sin x)(-\sin x)-\left(\cos x\right)\left(\cos x\right)}{{(1+\sin x)}^2}$
$=\frac{-\sin x-{\sin }^{2}x-{\cos }^{2}x}{{(1+\sin x)}^2}$
$=\frac{-\sin x-({\sin }^{2}x+{\cos }^{2}x)}{{(1+\sin x)}^2}$
$=\frac{-\sin x-1}{{(1+\sin x)}^2}$
$=\frac{-(1+\sin x)}{{(1+\sin x)}^2}=\frac{-1}{(1+\sin x)}$