Differentiation

Calculus Tutorial

Differentiation

Intro

Differentiation of a function y = f(x)^g(x)

Sample Problem

Problem : Find the differentiation of a function defined as y = f(x)^g(x). Eg : x^x, sinx^cosx etc.

Solution

Though we can differentiate the above function by taking log on both side and then applying product rule of differentiation, there is a 2nd method which uses the formula of differentiation of power function i.e. x^n and exponential function i.e. a^x.

We will do it in two steps :

Step 1. Differentiate f(x)^g(x) keeping f(x) as constant and hence function would be an exponential function. So the result would be f(x)^g(x)*ln(f(x))*g'(x) ——– (1)

Step 2. Differentiate f(x)^g(x) keeping g(x) as constant and hence function would be a power function. So the result would be g(x)*f(x)^{g(x)-1}*f'(x) ———— (2)

Now the final answer will be sum of expressions (1) & (2).

Eg. 1. Let y = x^x
dy/dx = x^(x-1).1 + x^x*ln(x)*1

2. y = sinx^cosx
dy/dx = cosx*sinx^(cosx-1)*(cosx) + sinx^cosx*ln(sinx)*(-sinx)



About The Author

Math Expert
My expertise are in teaching mathematics to secondary school level students preparing for IITJEE in India. IITJEE is toughest entrance examination in India at secondary school level and students clearing this entrance test are recognized world wide for their prowess in Physics, Chemistry, and Mathem...
11 Subjects
KnowRo Tutor
2 Tutorials
Chennai, TN
Get Tutoring Info

Suggested Tutors for Calculus Help

Abdul B

Arlington, VA

Math And Science Expert

Siddarth C

Vienna, VA

All Subjects

Emily V

Alexandria, VA

Versatile And Flexible Tutor

Maxine J

Bethesda, MD

Any Topic, Any Language

^