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.

Alina K. GCSE Maths tutor, GCSE Physics tutor, GCSE Chemistry tutor, ...

2 years ago

Answered by Alina, a GCSE Maths tutor with MyTutor

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


£20 /hr

Eszter H.

Degree: Modern Languages (Bachelors) - Durham University

Subjects offered:Maths, History+ 6 more

English and World Literature
English Literature
English Language
.MLAT (Modern Languages)

“I am a Modern Languages (German and Spanish) student at Durham University. I believe that the knowledge of foreign languages is becoming more and more important, thus I am devoted to help every student who want to develop their German...”

£18 /hr

Iebad A.

Degree: Aerospace Engineering (Masters) - Sheffield University

Subjects offered:Maths, Italian


“About me: Hello, my name is Iebad and I am an aerospace engineering student at The University of Sheffield. As an engineer, maths is the key to unlock all the solutions to our problems in this field. Since the early years I had a broad...”

Sahil M. A Level Biology tutor, GCSE Biology tutor, A Level Chemistry...
£20 /hr

Sahil M.

Degree: Medicine (Bachelors) - Manchester University

Subjects offered:Maths, Spanish+ 3 more

-Medical School Preparation-

“Hello! My name is Sahil and I am currently a medical student at the University of Manchester. I am really looking forward to tutoring you in Spanish, Chemistry, Biology or Maths as well as trying to help you during the difficult proce...”

About the author

Alina K.

Currently unavailable:

Degree: Mathematics (Bachelors) - Imperial College London University

Subjects offered:Maths, Russian+ 4 more

Further Mathematics

“I have always loved tutoring students and helping them to get a good understanding in their subject, specifically the Sciences and Mathematics. I have helped with A-level, IB and GCSE MathematicsI took GCSE Mathematics, Advanced Math...”

MyTutor guarantee

You may also like...

Other GCSE Maths questions

Integrate ∫_(-1)^1 3/√(x+2) dx using the substitution u x+2

How would I go about drawing a graph if i had no idea how to start?

There are "n" sweets in a bag, six are orange and the rest are yellow. If you take a random sweet from the bag and eat it. Then take at random another one and eat it. The probability of eating two orange sweets is 1/3. Show that n²-n-90=0.

Solve 9x-4=7(x+2)

View GCSE Maths tutors

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