Give the indicated derivative: f(x) = `(x*sin x + x^2)/(x^2+1)`

Expert Answers

An illustration of the letter 'A' in a speech bubbles

The derivative of f(x) = `(x*sin x + x^2)/(x^2+1)` is required. Use the quotient rule.

f'(x) = `((x*sin x + x^2)'*(x^2+1) - (x*sin x + x^2)(x^2+1)')/(x^2 + 1)^2`

=>` ((sin x+x*cos x+2x)*(x^2+1) - (x*sin x + x^2)*2x)/(x^2 + 1)^2`

=> `(x^2*sin x+x^3cos x+2x^3 +sin x+x*cos x+2x - (2x^2*sin x + 2x^3))/(x^2 + 1)^2`

=> `(x^3*cos x + sin x + x*cos x + 2x - x^2*sin x)/(x^2 + 1)^2`

=> `(cos x(x^3 + x) + sin x(1 -x^2) + 2x)/(x^2 + 1)^2`

The derivative of f(x) = `(x*sin x + x^2)/(x^2+1)` is f'(x) = `(cos x(x^3 + x) + sin x(1 -x^2) + 2x)/(x^2 + 1)^2`

Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial