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)#