Gujarat BoardEnglish MediumSTD 11 ScienceMATHSDerivatives2 Marks
Question
Differentiate the following function with respect to $\text{x}:$$\text{x}^\text{n}\log_\text{a}\text{x}$
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Answer
Let $\text{u}=\text{x}^\text{n};\text{v}=\log_\text{a}\text{x}=\frac{\log\text{x}}{\log\text{a}}$Then, $\text{u}'=\text{nx}^{\text{n}-1};\text{v}'=\frac{1}{\text{x}\log\text{a}}$
Using the product rule:
$\frac{\text{d}}{\text{dx}}(\text{uv})=\text{uv}'+\text{vu}'$
$\frac{\text{d}}{\text{dx}}(\text{x}^\text{n}\log_\text{a}\text{x})=\text{x}^\text{n}.\frac{1}{\text{x}\log\text{a}}+\log_\text{a}\text{x}(\text{nx}^{\text{n}-1})$
$=\text{x}^{\text{n}-1}\frac{1}{\log\text{a}}+\log_\text{a}\text{x}(\text{nx}^{\text{n}-1})$
$=\text{x}^{\text{n}-1}\Big(\frac{1}{\log\text{a}}+\text{n}\log_{\text{a}}\text{x}\Big)$
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