How do you differentiate x^x?

There are two ways we can find the derivative of x^x. It's important to notice that this function is neither a power function of the form x^k nor an exponential function of the form b^x, so we can't use the differentiation formulas for either of these cases directly.

(i) Let y=x^x, and take logarithms of both sides of this equation: ln(y)=ln(x^x). Using properties of logarithmic functions, we can rewrite this as ln(y)=x.ln(x). Then differentiating both sides with respect to x, and using the chain rule on the LHS and product rule on the RHS, gives 1/y.dy/dx=ln(x)+1. Rearranging, we have dy/dx=y.(ln(x)+1). That is, dy/dx=x^x(ln(x)+1).

(ii) Write x^x=e^(ln(x^x))=e^(x.ln(x)), using the properties of the exponential and logarithmic functions. Now, d/dx(x.ln(x))=ln(x)+1 by the product rule. Hence, d/dx(e^(x.ln(x)))=(ln(x)+1).(e^(x.ln(x)) by the chain rule, and using the fact that the derivative of e^[f(x)]=f'(x).e^[f(x)] for any differentiable function f(x). Finally, rewriting e^(x.ln(x)) as x^x gives d/dx(x^x)=x^x.(ln(x)+1), as with the first method.

Louis S. A Level Maths tutor, A Level Further Mathematics  tutor, A L...

2 years ago

Answered by Louis, an A Level Further Mathematics tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Emily-Louisa S.

Degree: Mathematics (Masters) - Exeter University

Subjects offered: Further Mathematics , Maths

Further Mathematics

“I am in my first year of my mathematics degree at the University of Exeter and one of my favourite things about maths is how it can appear in unexpected situations. For example, in cryptography (the writing and solving of codes of whi...”

£24 /hr

Joe C.

Degree: Mathematics (Masters) - Bristol University

Subjects offered: Further Mathematics , Maths

Further Mathematics

“I'm a third year student at Bristol University studying a Masters Degree in Mathematics. I try to make my tutorials engaging, and tailor them to your individual needs so that we are working towards the grade you aspire to achieve! I'v...”

£24 /hr

Joe B.

Degree: Mathematics G100 (Bachelors) - Bath University

Subjects offered: Further Mathematics , Maths+ 4 more

Further Mathematics
-Personal Statements-

“About Me Hi, I'm Joe, a first year mathematics student from Luton studying at Bath University. I am an accomplished mathematician and economist, having achieved A* grades in A Level Maths, Further Maths and Economics in June 2016. As ...”

About the author

£30 /hr

Louis S.

Degree: Mathematics (Bachelors) - Cambridge University

Subjects offered: Further Mathematics , Physics+ 6 more

Further Mathematics
Extended Project Qualification
-Personal Statements-
-Oxbridge Preparation-

“ second year undergraduate at the University of Cambridge, studying for a B.A. in Mathematics, having received A*s in A-Level Mathematics, Further Mathematics (Edexcel) and Physics (AQA)”

You may also like...

Other A Level Further Mathematics questions

Express cos(4x) in terms of powers of cos(x)

How do you plot a complex number in an Argand diagram?

How does proof by mathematical induction work?

How do I solve a simultaneous equation with more unknowns than equations?

View A Level Further Mathematics tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss