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Quotient Rule

Quotient Rule Examples

Quotient Rule of Differentiation

The Quotient Rule of Differentiation states that when you seek to differentiate a function comprised of a quotient of other functions, the result is as follows:

Given: \(f(x) = \frac{u(x)}{v(x)}\)

The Quotient Rule is: \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\)

Example

Differentiate \(f(x) = \frac{sin(x)}{x^2+1}\)

Here \(u(x) = sin(x)\) and \(v(x) = x^2 + 1\)

Thus, \(f'(x) = \frac{(cos(x))(x^2 + 1) - (sin(x))(2x)}{(x^4 + 2x^2 + 1)}\)

Practice Problems

  1. \(f(x) = \frac{2x^2 + 1}{3x - 2}\)
  2. \(f(x) = \frac{\sin(x)}{\cos(x)} = \tan(x)\)
  3. \(f(x) = \frac{x - 1}{e^x}\)
  4. \(f(x) = \frac{\sin(x)}{\cos(3x - 1)}\)
  5. \(f(x) = \frac{x^2 + 1}{x^2 - 1}\)
  6. \(f(x) = \frac{\cos(x)}{\sin(x)}\)
  7. \(f(x) = \frac{(x^2 - 1)}{x^2 + 1}\)
  8. \(f(x) = \frac{(3x + \sin(x))}{e^x + \ln(x)}\)
  9. \(f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}\)
  10. \(f(x) = \frac{x - 1}{x + 1}\)
  11. \(f(x) = \frac{\sin(2x) - e^x}{\cos(2x)}\)
  12. \(f(x) = \frac{\ln(3x + 1)}{\cos(-3x^2)}\)
  13. \(f(x) = \frac{3x^2 + 2x^3}{\sin(3x) - \cos(2x)}\)
  14. \(f(x) = \frac{(\sin(x) + x)^2}{x^2}\)
  15. \(f(x) = \frac{2x - \sin(x)}{2 - \ln(x)}\)
  16. \(f(x) = \frac{e^x - \ln(xe)}{\sin(x) + x^2}\)
  17. \(f(x) = \frac{\sin(-3x/2)}{\frac{x^2-1}{2x+1} \cos(x)}\)
  18. \(f(x) = \frac{2x^2 - 5x}{\sin(-4x) \cos(-4x)}\)
  19. \(f(x) = \frac{(2x + 1)(3 \sin(x))}{x^2 + 1}\)
  20. \(f(x) = \frac{(2x^3)(\cos(x))}{\sin(x)(x + x^2)}\)