Find derivative of $(a x + b)^{n} (c x + d)^{n}$

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Solution

$f (x) = (a x + b)^{n} (c x + d)^{n}$

$u = (a x + b)^{n} \Rightarrow u^{'} = a n (a x + b)^{n - 1}$

$v = (c x + d)^{n} \Rightarrow v^{'} = c n (c x + d)^{n - 1}$

$f^{'} (x) = (u v)^{'}$

$= u v^{'} + v u^{'}$

$= (a x + b)^{n} c n (c x + b)^{n - 1} + (c x + b)^{n} a n (a x + b)^{n - 1}$

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Find derivative of (ax+b)n(cx+d)n

f(x)=(ax+b)n(cx+d)nu=(ax+b)n⇒u′=an(ax+b)n−1v=(cx+d)n⇒v′=cn(cx+d)n−1f′(x)=(uv)′=uv′+vu′=(ax+b)ncn(cx+b)n−1+(cx+b)nan(ax+b)n−1

Find derivative of $(a x + b)^{n} (c x + d)^{n}$

$f (x) = (a x + b)^{n} (c x + d)^{n}$

$u = (a x + b)^{n} \Rightarrow u^{'} = a n (a x + b)^{n - 1}$

$v = (c x + d)^{n} \Rightarrow v^{'} = c n (c x + d)^{n - 1}$

$f^{'} (x) = (u v)^{'}$

$= u v^{'} + v u^{'}$

$= (a x + b)^{n} c n (c x + b)^{n - 1} + (c x + b)^{n} a n (a x + b)^{n - 1}$