Find the derivatives of the following :

(ax+b)ⁿ(cx+ d)ⁿ

Answer

Let y = (ax+b)ⁿ(cx+ d)ⁿ
differentiating both sides w.r.t. x, we get
$space space space space space space space space space dy over dx equals straight d over dx left square bracket left parenthesis ax plus straight b right parenthesis to the power of straight n space left parenthesis cx plus space straight d right parenthesis to the power of straight n right square bracket$
 $space space space space space space space space space space space space space space space space space equals space space space space left parenthesis ax space plus straight b right parenthesis to the power of straight n straight n left parenthesis cx plus straight d right parenthesis to the power of straight n minus 1 space end exponent straight c space plus space left parenthesis cx plus straight d right parenthesis to the power of straight n space straight d over dx left parenthesis ax plus straight b right parenthesis to the power of straight n$
 $space space space space space space space equals space space space space left parenthesis ax plus straight b right parenthesis to the power of straight n straight n left parenthesis cx plus straight d right parenthesis to the power of straight n minus 1 end exponent straight c plus left parenthesis cx plus straight d right parenthesis to the power of straight n space straight n space left parenthesis ax plus straight b right parenthesis to the power of straight n minus 1 end exponent a$
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 $space space space space space space space space space space space space equals space straight n open parentheses ax plus straight b close parentheses to the power of straight n minus 1 end exponent open parentheses cx plus straight d close parentheses to the power of straight n minus 1 end exponent open parentheses 2 ax plus bc plus ad close parentheses$