I would like to know the derivatives of the following function:
$ y = x^e*e^x$
At first sight it looks like the product rule should be used and so one would get $e*x^{e-1}*e^x+x^e*e^x$. Is this correct?
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1Yes, it is correct. – Marra Apr 24 '13 at 01:01
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1Does $\ast$ denote product or convolution? – Qiaochu Yuan Apr 24 '13 at 01:03
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just product... – Apr 24 '13 at 01:06
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1Sure, it's fine. – André Nicolas Apr 24 '13 at 01:14
1 Answers
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Yes indeed, your answer is correct; you need only the product rule:
$$ \begin{align} y & = x^e\cdot e^x \\ \\ y' & = e \cdot x^{e-1}\cdot e^x + x^e\cdot e^x \tag{we can "factor" this}\\ \\ & = {\bf e} \cdot \color{blue}{\bf x^{e - 1} \cdot e^x} + {\bf x} \cdot \color{blue}{\bf x^{e - 1} \cdot e^x} \\ \\ & = {\bf(e + x)}\cdot \color{blue}{\bf x^{e-1}\cdot e^x} \qquad\qquad\qquad\qquad\tag{factored} \end{align} $$
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