What is the derivative of $x^e$ ?

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Answer 1 · 2016-01-31T00:08:22

$d/dx x^e = ex^(e-1)$

If $k$ is a constant, then the power rule states that

$d/dx x^k = kx^(k-1)$

$e$ is no different.

$d/dx x^e = ex^(e-1)$

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What follows is a short proof of the power rule for any real constant using implicit differentiation and the derivative $d/dxln(x) = 1/x$ .

Let $k in RR$ be constant, and let $y = x^k$

Then $ln(y) = ln(x^k) = kln(x)$

Differentiating, we have $d/dxlny = d/dxkln(x)$

$=> 1/ydy/dx = k/x$

$=> dy/dx = ky/x = kx^k/x = kx^(k-1)$

$:. d/dxx^k = kx^(k-1)$

What is the derivative of x^ex^e?