Implicit and Logarithmic Differentiation

Implicit Differentiation

  • Definition: Implicit differentiation is a technique used to find derivatives of functions that are not explicitly solved for one variable in terms of another.
  • Example:
    • Differentiate the implicit equation:
      d(6+(−))=4d(6 + (-)) = 4
      dy/dx=y′dy/dx = y'
  • General Rule for Implicit Differentiation: When differentiating an equation with respect to $x$, treat $y$ as a function of $x$ (i.e., use the chain rule).

Composite Functions

  • Definition: A composite function is a function formed by applying one function to the results of another.
  • Example:
    • If $g(x)$ is the inner function and $f$ is the outer function, then $y = f(g(x))$.
  • Differentiation of Composite Functions: Generally,
    dy/dx=f′(g(x))imesg′(x)dy/dx = f'(g(x)) imes g'(x)
  • Relate this to the product of the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Derivatives of Logarithmic Functions

  • Basic Rule:
    • For the function $y = ext{log}_b{x}$, the derivative is given by
      dy/dx=1ximesextln(b)dy/dx = \frac{1}{x imes ext{ln}(b)}
  • Example of Differentiation:
    • Differentiate the function $y = ext{log}_b{x}$:
      dy/dx=1ximesextln(b)dy/dx = \frac{1}{x imes ext{ln}(b)}
    • In the case where $b=e$, the derivative simplifies to
      dy/dx=1xdy/dx = \frac{1}{x}

Logarithmic Rule

  • Usage: Helpful for functions involving logarithms.
  • Example: Differentiate the function $f(x) = x imes ext{ln}(x)$.
    • Apply Product Rule:
      f′(x)=x′imesextln(x)+ximes(extln(x))′f'(x) = x' imes ext{ln}(x) + x imes ( ext{ln}(x))'
      =extln(x)+ximes1x= ext{ln}(x) + x imes \frac{1}{x}
      =extln(x)+1= ext{ln}(x) + 1

Differentiating Natural Logarithm

  • Example Function:
    f(x)=extln(1x)f(x) = ext{ln}(\frac{1}{x})
  • Finding Domain (Df):
    • The restrictions for the function include:
    • |x| > 0
    • Thus the domain is (−ext∞,0)⋃(0,ext∞)(- ext{∞}, 0) \bigcup (0, ext{∞})
  • Derivative:
    • For x > 0,
      (extln(x))′=1x( ext{ln}(x))' = \frac{1}{x}
    • For x < 0,
      (extln(−x))′=−1−x=1x( ext{ln}(-x))' = \frac{-1}{-x} = \frac{1}{x}

Logarithmic Differentiation

  • Process: Useful when differentiating complex functions.
  • Example:
    • Differentiate y=xxy = x^x.
    • Transform: Take the natural logarithm on both sides:
      extln(y)=extln(xx)ext{ln}(y) = ext{ln}(x^x)
      extln(y)=ximesextln(x)ext{ln}(y) = x imes ext{ln}(x)
    • Differentiate implicitly with respect to xx:
      ddx(extln(y))=dydximes1y=ddx(ximesextln(x))\frac{d}{dx}( ext{ln}(y)) = \frac{dy}{dx} imes \frac{1}{y} = \frac{d}{dx}(x imes ext{ln}(x))
    • Apply product rule:
      dydximes1y=(1imesextln(x))+(ximes1x)\frac{dy}{dx} imes \frac{1}{y} = (1 imes ext{ln}(x)) + (x imes \frac{1}{x})
      =extln(x)+1= ext{ln}(x) + 1
    • Finally, re-arranging gives:
      dydx=y(extln(x)+1)\frac{dy}{dx} = y( ext{ln}(x) + 1).

Conclusion of Logarithmic Differentiation

  • Implication: Logarithmic differentiation provides an efficient way to differentiate products/powers of variables that may prove difficult to handle otherwise.
  • Key Takeaway: Use logarithmic differentiation when facing complex variable expressions, particularly when both bases and exponents involve the variable itself.