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