Derivatives of Logarithmic and Exponential Functions

Introduction

• Discuss derivatives of logarithmic and exponential functions
• Graph of exponential function (in blue)
• Graph of logarithmic function (in red)
• Logs and exponentials are inverse functions

Derivative of Exponential Functions

• Derivative of ( b^x ) is ( b^x \ln b )
  • "b" must be greater than 0 to avoid complex numbers
• Special case: ( e^x ) derivative is ( e^x ) because ( \ln e = 1 )
• Chain rule applies for ( b^{g(x)} ):
  • Derivative: ( b^{g(x)} \ln b \cdot g'(x) )

Limit Definition of Derivative

• Use limit definition to derive ( b^x \ln b )
  • ( \lim_{h \to 0} \frac{b^{x+h} - b^x}{h} )
  • Results in the derivative rule: ( b^x \ln b )

Derivative Rules Clarification

• Exponential rule: number to a variable (e.g., ( b^x ))
• Power rule: variable to a number (e.g., ( x^n ))
• Do not mix these rules

Derivative of Logarithmic Functions

• ( y = \log_b{x} \rightarrow \frac{d}{dx}\log_b{x} = \frac{1}{x \ln b} )
  • Special case: ( \ln x ) derivative is ( \frac{1}{x} )
• Chain rule application for ( \log_b{g(x)} ):
  • Derivative: ( \frac{1}{g(x) \ln b} \cdot g'(x) )

Proof Through Implicit Differentiation

• Start with ( y = \log_b x ), which implies ( b^y = x )
• Use implicit differentiation to find derivative
• Result: ( \frac{d}{dx}\log_b{x} = \frac{1}{x \ln b} )

Examples and Applications

• Discussed several examples applying these rules
  • Included examples with chain rule, product rule, and quotient rule
  • Simplification using properties of logarithms (e.g., splitting products and quotients)

Logarithmic Differentiation

• Used when both base and exponent are variables
• Example: ( y = x^x )
  • Take natural log of both sides
  • Apply implicit differentiation
  • Replace y with ( x^x ) in derivative expression
• Result: ( \frac{d}{dx}(x^x) = x^x (\ln x + 1) )

More Examples with Logarithmic Differentiation

• Examples of functions like ( x^{\sin{\pi x}} )
  • Take natural log, use properties, differentiate implicitly
  • Replace y with its original expression at the end

Conclusion

• Proper distinction between rules is crucial
• Use logarithmic differentiation when necessary
• Apply chain rule, product rule, and quotient rule appropriately
• Simplification can aid in deriving derivatives efficiently