Study Notes on Derivatives of Logarithmic and Exponential Functions

Derivative of Logarithms

  • ddxln⁡(x)=1x\frac{d}{dx} \ln(x) = \frac{1}{x}

Chain Rule Version for Logarithmic Derivatives

  • ddxln⁡(f(x))=f′(x)f(x)\frac{d}{dx} \ln(f(x)) = \frac{f'(x)}{f(x)}

Derivative involving a logarithm of a power

  • ln⁡(xn)=nln⁡(x)\ln(x^n) = n\ln(x)

  • ddxln⁡(xn)=n⋅1x\frac{d}{dx} \ln(x^n) = n \cdot \frac{1}{x}

Second Derivative and High-Degree Functions

  • For functions of the form g(x)=ln⁡(f(x))g(x) = \ln(f(x)): apply product or quotient rule as needed.

Application of Derivative Rules

  • Simplifications such as ln⁡(x2+1)=12ln⁡(x2+1)\ln(\sqrt{x^2 + 1}) = \frac{1}{2} \ln(x^2 + 1) can facilitate differentiation.