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