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