MYTUTOR SUBJECT ANSWERS

615 views

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

453 SUBJECT SPECIALISTS

£24 /hr

Octavia C.

Degree: History (Bachelors) - Durham University

Subjects offered: Maths, Politics+ 2 more

Maths
Politics
History
English Literature

“About Me:  My name is Octavia Cogher and I am in my second year studying history at Durham University. I am interested in a broad range of topics and periods, but particularly in twentieth-century political history. Last year, I tuto...”

£18 /hr

Amelia F.

Degree: Social Anthropology MA (Bachelors) - Edinburgh University

Subjects offered: Maths, Extended Project Qualification+ 1 more

Maths
Extended Project Qualification
English Literature

“I am a Social Anthropology student at the University of Edinburgh. I have always had a passion for learning and for helping others and I am hoping that I will be able to give my love of education back to my students. I am very patient...”

MyTutor guarantee

£30 /hr

Araba S.

Degree: Business & Marketing (Masters) - Warwick University

Subjects offered: Maths, Further Mathematics

Maths
Further Mathematics

“About me: I am a Chemical Engineering graduate currently studying a Masters in Business at the Warwick Business School. Maths is a subject I thoroughly enjoyed at school and nothing brought me more satisfaction than breaking down othe...”

MyTutor guarantee

About the author

Alina K.

Currently unavailable:

Degree: Mathematics (Bachelors) - Imperial College London University

Subjects offered: Maths, Russian+ 4 more

Maths
Russian
Physics
Further Mathematics
Economics
Chemistry

“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

Work out the value of 125 to the power of -2/3.

Solve the simultaneous equations: 3a + 2b = 17 and 4a - b = 30

Solve these equations simultaneously: (1) 5x - 10z = -45 and (2) 9x = -5z + 80

Factorise y^2 + 7y + 6

View GCSE Maths tutors

Cookies:

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

mtw:mercury1:status:ok