Derivatives - Examples of Quotient Rule

  • The quotient rule is used to find the derivative of a function that is expressed as the quotient of two other functions
  • It is given by the formula: [ \frac{{d}}{{dx}}\left(\frac{{f(x)}}{{g(x)}}\right) = \frac{{g(x)f’(x) - f(x)g’(x)}}{{[g(x)]^2}} ]
  • Let’s look at some examples to understand how to apply the quotient rule

Example 1

Find the derivative of the function (\displaystyle f(x) = \frac{{x^2}}{{x+2}})

  • Step 1: Identify (f(x)) and (g(x))
    • (f(x) = x^2)
    • (g(x) = x+2)
  • Step 2: Find (f’(x)) and (g’(x))
    • (f’(x) = 2x)
    • (g’(x) = 1)
  • Step 3: Apply the quotient rule formula
    • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{f(x)}}{{g(x)}}\right) = \frac{{g(x)f’(x) - f(x)g’(x)}}{{[g(x)]^2}})
    • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{x^2}}{{x+2}}\right) = \frac{{(x+2)(2x) - (x^2)(1)}}{{(x+2)^2}})

Example 2

Find the derivative of the function (\displaystyle f(x) = \frac{{5x}}{{3x-1}})

  • Step 1: Identify (f(x)) and (g(x))
    • (f(x) = 5x)
    • (g(x) = 3x-1)
  • Step 2: Find (f’(x)) and (g’(x))
    • (f’(x) = 5)
    • (g’(x) = 3)
  • Step 3: Apply the quotient rule formula
    • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{f(x)}}{{g(x)}}\right) = \frac{{g(x)f’(x) - f(x)g’(x)}}{{[g(x)]^2}})
    • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{5x}}{{3x-1}}\right) = \frac{{(3x-1)(5) - (5x)(3)}}{{(3x-1)^2}})

Example 3

Find the derivative of the function (\displaystyle f(x) = \frac{{2x^3}}{{x^2-4}})

  • Step 1: Identify (f(x)) and (g(x))
    • (f(x) = 2x^3)
    • (g(x) = x^2-4)
  • Step 2: Find (f’(x)) and (g’(x))
    • (f’(x) = 6x^2)
    • (g’(x) = 2x)
  • Step 3: Apply the quotient rule formula
    • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{f(x)}}{{g(x)}}\right) = \frac{{g(x)f’(x) - f(x)g’(x)}}{{[g(x)]^2}})
    • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{2x^3}}{{x^2-4}}\right) = \frac{{(x^2-4)(6x^2) - (2x^3)(2x)}}{{(x^2-4)^2}})
  1. Application of Quotient Rule - Examples continued
  • Example 4: Find the derivative of the function (\displaystyle f(x) = \frac{{4x^2+5}}{{3x^3-2}})
    • Step 1: Identify (f(x)) and (g(x))
      • (f(x) = 4x^2+5)
      • (g(x) = 3x^3-2)
    • Step 2: Find (f’(x)) and (g’(x))
      • (f’(x) = 8x)
      • (g’(x) = 9x^2)
    • Step 3: Apply the quotient rule formula
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{f(x)}}{{g(x)}}\right) = \frac{{g(x)f’(x) - f(x)g’(x)}}{{[g(x)]^2}})
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{4x^2+5}}{{3x^3-2}}\right) = \frac{{(3x^3-2)(8x) - (4x^2+5)(9x^2)}}{{(3x^3-2)^2}})
  1. Properties of Derivatives - Constant Multiple Rule
  • The constant multiple rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.
  • It can be expressed as:
    • (\displaystyle \frac{{d}}{{dx}}(cf(x)) = c \cdot \frac{{d}}{{dx}}f(x))
    • Where (c) is a constant.
  1. Example of Constant Multiple Rule
  • Example: Find the derivative of the function (\displaystyle f(x) = 7x^2)
    • Step 1: Identify (f(x))
      • (f(x) = 7x^2)
    • Step 2: Apply the constant multiple rule
      • (\displaystyle \frac{{d}}{{dx}}(7x^2) = 7 \cdot \frac{{d}}{{dx}}(x^2) = 7 \cdot 2x = 14x)
  1. Properties of Derivatives - Sum/Difference Rule
  • The sum/difference rule states that the derivative of the sum or difference of two functions is equal to the sum or difference of their derivatives.
  • It can be expressed as follows:
    • (\displaystyle \frac{{d}}{{dx}}(f(x) \pm g(x)) = \frac{{d}}{{dx}}f(x) \pm \frac{{d}}{{dx}}g(x))
  1. Example of Sum/Difference Rule
  • Example: Find the derivative of the function (\displaystyle f(x) = 2x^3 + 4x^2)
    • Step 1: Identify (f(x))
      • (f(x) = 2x^3 + 4x^2)
    • Step 2: Apply the sum rule
      • (\displaystyle \frac{{d}}{{dx}}(2x^3 + 4x^2) = \frac{{d}}{{dx}}(2x^3) + \frac{{d}}{{dx}}(4x^2) = 6x^2 + 8x)
  1. Properties of Derivatives - Power Rule
  • The power rule is used to find the derivative of a function that is raised to a constant power.
  • It is given by the formula:
    • (\displaystyle \frac{{d}}{{dx}}(x^n) = nx^{n-1})
    • Where (n) is a constant.
  1. Example of Power Rule
  • Example 1: Find the derivative of the function (\displaystyle f(x) = 3x^4)
    • Step 1: Identify (f(x))
      • (f(x) = 3x^4)
    • Step 2: Apply the power rule
      • (\displaystyle \frac{{d}}{{dx}}(3x^4) = 4 \cdot 3x^{4-1} = 12x^3)
  1. Example of Power Rule (continued)
  • Example 2: Find the derivative of the function (\displaystyle f(x) = \frac{{x^2}}{{4}})
    • Step 1: Identify (f(x))
      • (f(x) = \frac{{x^2}}{{4}})
    • Step 2: Apply the power rule
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{x^2}}{{4}}\right) = \frac{{2x^{2-1}}}{{4}} = \frac{{2x}}{4} = \frac{{x}}{2})
  1. Chain Rule
  • The chain rule is used to find the derivative of a composite function.
  • It is given by the formula:
    • (\displaystyle \frac{{d}}{{dx}}(f(g(x))) = f’(g(x)) \cdot g’(x))
  1. Example of Chain Rule
  • Example: Find the derivative of the function (\displaystyle f(x) = (2x+1)^3)
    • Step 1: Identify (f(x)) and (g(x))
      • (f(x) = x^3)
      • (g(x) = 2x+1)
    • Step 2: Find (f’(x)) and (g’(x))
      • (f’(x) = 3x^2)
      • (g’(x) = 2)
    • Step 3: Apply the chain rule formula
      • (\displaystyle \frac{{d}}{{dx}}((2x+1)^3) = f’(g(x))\cdot g’(x) = 3(2x+1)^2 \cdot 2) Sure! Here are slides 21 to 30 on the topic of “Derivatives - Examples of Quotient Rule”:
  1. Quotient Rule - Recap
  • The quotient rule is used to find the derivative of a function that is expressed as the quotient of two other functions.
  • It can be expressed as:
    • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{f(x)}}{{g(x)}}\right) = \frac{{g(x)f’(x) - f(x)g’(x)}}{{[g(x)]^2}})
  1. Example 1 Find the derivative of the function (\displaystyle f(x) = \frac{{x^3}}{{x+1}})
  • Solution:
    • Step 1: Identify (f(x)) and (g(x))
      • (f(x) = x^3)
      • (g(x) = x+1)
    • Step 2: Find (f’(x)) and (g’(x))
      • (f’(x) = 3x^2)
      • (g’(x) = 1)
    • Step 3: Apply the quotient rule formula
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{x^3}}{{x+1}}\right) = \frac{{(x+1)(3x^2) - (x^3)(1)}}{{(x+1)^2}})
  1. Example 2 Find the derivative of the function (\displaystyle f(x) = \frac{{2x}}{{x^2+3}})
  • Solution:
    • Step 1: Identify (f(x)) and (g(x))
      • (f(x) = 2x)
      • (g(x) = x^2+3)
    • Step 2: Find (f’(x)) and (g’(x))
      • (f’(x) = 2)
      • (g’(x) = 2x)
    • Step 3: Apply the quotient rule formula
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{2x}}{{x^2+3}}\right) = \frac{{(x^2+3)(2) - (2x)(2x)}}{{(x^2+3)^2}})
  1. Example 3 Find the derivative of the function (\displaystyle f(x) = \frac{{3x^4+2x^3}}{{x^2+1}})
  • Solution:
    • Step 1: Identify (f(x)) and (g(x))
      • (f(x) = 3x^4+2x^3)
      • (g(x) = x^2+1)
    • Step 2: Find (f’(x)) and (g’(x))
      • (f’(x) = 12x^3+6x^2)
      • (g’(x) = 2x)
    • Step 3: Apply the quotient rule formula
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{3x^4+2x^3}}{{x^2+1}}\right) = \frac{{(x^2+1)(12x^3+6x^2) - (3x^4+2x^3)(2x)}}{{(x^2+1)^2}})
  1. Example 4 Find the derivative of the function (\displaystyle f(x) = \frac{{\sin(x)}}{{\cos(x)}})
  • Solution:
    • Step 1: Identify (f(x)) and (g(x))
      • (f(x) = \sin(x))
      • (g(x) = \cos(x))
    • Step 2: Find (f’(x)) and (g’(x))
      • (f’(x) = \cos(x))
      • (g’(x) = -\sin(x))
    • Step 3: Apply the quotient rule formula
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{\sin(x)}}{{\cos(x)}}\right) = \frac{{(\cos(x))(\sin(x)) - (\sin(x))(-\sin(x))}}{{(\cos(x))^2}})
  1. Example 5 Find the derivative of the function (\displaystyle f(x) = \frac{{2x^2-3x-1}}{{x+2}})
  • Solution:
    • Step 1: Identify (f(x)) and (g(x))
      • (f(x) = 2x^2-3x-1)
      • (g(x) = x+2)
    • Step 2: Find (f’(x)) and (g’(x))
      • (f’(x) = 4x-3)
      • (g’(x) = 1)
    • Step 3: Apply the quotient rule formula
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{2x^2-3x-1}}{{x+2}}\right) = \frac{{(x+2)(4x-3) - (2x^2-3x-1)(1)}}{{(x+2)^2}})
  1. Example 6 Find the derivative of the function (\displaystyle f(x) = \frac{{\sqrt x}}{{x+1}})
  • Solution:
    • Step 1: Identify (f(x)) and (g(x))
      • (f(x) = \sqrt x)
      • (g(x) = x+1)
    • Step 2: Find (f’(x)) and (g’(x))
      • (f’(x) = \frac{{1}}{{2\sqrt x}})
      • (g’(x) = 1)
    • Step 3: Apply the quotient rule formula
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{\sqrt x}}{{x+1}}\right) = \frac{{(x+1)\frac{{1}}{{2\sqrt x}} - (\sqrt x)(1)}}{{(x+1)^2}})
  1. Example 7 Find the derivative of the function (\displaystyle f(x) = \frac{{e^x}}{{1+e^x}})
  • Solution:
    • Step 1: Identify (f(x)) and (g(x))
      • (f(x) = e^x)
      • (g(x) = 1+e^x)
    • Step 2: Find (f’(x)) and (g’(x))
      • (f’(x) = e^x)
      • (g’(x) = e^x)
    • Step 3: Apply the quotient rule formula
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac{{e^x}}{{1+e^x}}\right) = \frac{{(1+e^x)(e^x) - (e^x)(e^x)}}{{(1+e^x)^2}})
  1. Example 8 Find the derivative of the function (\displaystyle f(x) = \frac{{\ln(x)}}{{x^2}})
  • Solution:
    • Step 1: Identify (f(x)) and (g(x))
      • (f(x) = \ln(x))
      • (g(x) = x^2)
    • Step 2: Find (f’(x)) and (g’(x))
      • (f’(x) = \frac{{1}}{{x}})
      • (g’(x) = 2x)
    • Step 3: Apply the quotient rule formula
      • (\displaystyle \frac{{d}}{{dx}}\left(\frac