Differentiate the following equation:
$(a x + b)^{n} (c x + d)^{m}$

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Solution

$= \frac{d}{d x} (a x + b)^{n} (c x + d)^{m}$
$= (a x + b)^{n} \frac{d}{d x} (c x + d)^{m} + (c x + d)^{m} \frac{d}{d x} (a x + b)^{n}$
$= (a x + b)^{n} [(n (c x + d)^{m - 1}) . c] + (c x + d)^{m} [(n (a x + d)^{n - 1}) . a]$
$= n (a x + b)^{n - 1} (c x + d)^{m - 1} [(a x + b) c + (c x + d) a])$

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Differentiate the following equation: (ax+b)n(cx+d)m

=ddx(ax+b)n(cx+d)m =(ax+b)nddx(cx+d)m+(cx+d)mddx(ax+b)n =(ax+b)n[(n(cx+d)m−1).c]+(cx+d)m[(n(ax+d)n−1).a] =n(ax+b)n−1(cx+d)m−1[(ax+b)c+(cx+d)a])

Differentiate the following equation:
$(a x + b)^{n} (c x + d)^{m}$

$= \frac{d}{d x} (a x + b)^{n} (c x + d)^{m}$
$= (a x + b)^{n} \frac{d}{d x} (c x + d)^{m} + (c x + d)^{m} \frac{d}{d x} (a x + b)^{n}$
$= (a x + b)^{n} [(n (c x + d)^{m - 1}) . c] + (c x + d)^{m} [(n (a x + d)^{n - 1}) . a]$
$= n (a x + b)^{n - 1} (c x + d)^{m - 1} [(a x + b) c + (c x + d) a])$