How to Differentiate E^X and X^X

Edited by Nathan Wong, Wes Alvaro, Peter, Teresa and 2 othersPin ItArticle EditDiscussAt A-Level you are told as a standard result that e^x differentiates to itself, but few people studying A-level Maths can actually prove it...
Steps
Differentiating e^x1Set up the equation: y=e^x

2Take the natural logarithm of both sides: ln y = x

3Differentiate implicitly: 1/y (dy/dx) = 1

4Multiply both sides by y: dy/dx = y

5Replace y with e^x: dy/dx = e^x


Differentiating x^x1Take the natural logarithm: ln y = xln x

2Differentiate implicitly with the product rule:

1/y (dy/dx) = x(1/x) + ln x1/y (dy/dx) = 1 + ln x3Multiply by y: dy/dx = y(1 + ln x)

4Replace y with x^x: dy/dx = x^x(1 + ln x)

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TipsRemember that ln x differentiates to 1/x.Practice the same technique to see if you can differentiate x^x to get x^x(1+ln x) using the product rule.This is possible question if applying to do Mathematics at Oxbridge so make sure you can do it if required.If you don't understand logs then, check out How to Understand Logarithms.
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Categories: Calculus
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