Differentiate the following function with respect to x:(ax+b)^n(c… - Vidyadip

Question

Differentiate the following function with respect to x:$(\text{ax}+\text{b})^\text{n}(\text{cx}+\text{d})^\text{m}$

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Answer

$(\text{ax}+\text{b})^\text{n}(\text{cx}+\text{d})^\text{m}$Let $\text{u}=(\text{ax}+\text{b})^\text{n},\text{v}=(\text{cx}+\text{d})^\text{m}$
Then, $\text{u}'=\text{na}(\text{ax}+\text{b})^{\text{n}-1},\text{v}'=\text{nc}(\text{cx}+\text{d})^{\text{m}-1}$
Using the product rule:
$\frac{\text{d}}{\text{dx}}(\text{uv})=\text{uv}'+\text{vu}'$
$\frac{\text{d}}{\text{dx}}[(\text{ax}+\text{b})^\text{n}(\text{cx}+\text{d})^\text{m}]=(\text{ax}+\text{b})^\text{n}\times\text{mc}(\text{cx}+\text{d})^{\text{m}-1}+(\text{cx}+\text{d})^\text{m}\times\text{na}(\text{ax}+\text{b})^{\text{n}-1}$
$\text{mc}(\text{ax}+\text{b})^\text{n}(\text{cx}+\text{d})^{\text{m}-1}+\text{na}(\text{cx}+\text{d})^\text{m}(\text{ax}+\text{b})^{\text{n}-1}$

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