MYTUTOR SUBJECT ANSWERS

831 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.

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

3 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

132 SUBJECT SPECIALISTS

£20 /hr

Mabast H.

Degree: Physics (Bachelors) - Imperial College London University

Subjects offered:Further Mathematics , Physics+ 3 more

Further Mathematics
Physics
Maths
Chemistry
-Personal Statements-

“Undergraduate Physicist with lots of experience in teaching in different fields and age groups, looking to instil a passion for learning into students.”

PremiumGiulio P. GCSE Maths tutor, A Level Maths tutor, A Level Economics tu...
£36 /hr

Giulio P.

Degree: Mathematics (Masters) - Bristol University

Subjects offered:Further Mathematics , Physics+ 3 more

Further Mathematics
Physics
Maths
Italian
Economics

“Exam Time! I can provide a rapid revision class in any maths module which will test your son or daughter in all the fundamentals”

£30 /hr

Tadas T.

Degree: MMathPhil Mathematics and Philosophy (Bachelors) - Oxford, St Anne's College University

Subjects offered:Further Mathematics , Maths+ 3 more

Further Mathematics
Maths
.MAT.
-Personal Statements-
-Oxbridge Preparation-

“University of Oxford Maths and Philosophy student happy to help students learn and stay motivated!”

About the author

Louis S.

Currently unavailable: for new students

Degree: Mathematics (Bachelors) - Cambridge University

Subjects offered:Further Mathematics , Physics+ 6 more

Further Mathematics
Physics
Maths
Extended Project Qualification
.STEP.
.MAT.
-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

A golf ball is hit from horizontal ground with speed 10 m/s at an angle of p degrees above the horizontal. The greatest height the golf ball reached above ground level is 1.22m. Model the golf ball as a particle and ignore air resistance. Find p.

Find the vector equation of the line of intersection of the planes 2x+y-z=4 and 3x+5y+2z=13.

Let P(z) = z⁴ + az³ + bz² + cz + d be a quartic polynomial with real coefficients. Let two of the roots of P(z) = 0 be 2 – i and -1 + 2i. Find a, b, c and d.

Prove that (AB)^-1 = B^-1 A^-1

View A Level Further Mathematics 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

mtw:mercury1:status:ok