Let f(x) be a differentiable function such that xf(x)+f(x^2)=2 for all x>0 find f'(1)?
in Calculus Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

Differentiate: xf'(x)+f(x)+2xf'(x^2)=0; f(x)+f(x^2)/x=2/x, so xf'(x)+2/x-f(x^2)/x+2xf'(x^2)=0.

Putting x=1, we have f'(1)+2-f(1)+2f'(1)=0. To find f(1), substitute x=1 in xf(x)+f(x^2)=2: f(1)+f(1)=2, so f(1)=1. Substitute for f(1) in the derivative: 3f'(1)+2-1=0, so f'(1)=-1/3.

by Top Rated User (642k points)

Related questions

1 answer
0 answers
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
82,892 questions
87,496 answers
1,965 comments
3,941 users