Summary of Derivatives of Exponential and Logarithmic Functions

On the basis of the assumption that the exponential function [latex]y=b^x, \, b>0[/latex] is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative.
We can use a formula to find the derivative of [latex]y=\ln x[/latex], and the relationship [latex]\log_b x=\dfrac{\ln x}{\ln b}[/latex] allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
Logarithmic differentiation allows us to differentiate functions of the form [latex]y=g(x)^{f(x)}[/latex] or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.

Key Equations

Inverse function theorem
[latex](f^{-1})^{\prime}(x)=\dfrac{1}{f^{\prime}(f^{-1}(x))}[/latex] whenever [latex]f^{\prime}(f^{-1}(x))\ne 0[/latex] and [latex]f(x)[/latex] is differentiable.
Power rule with rational exponents
[latex]\frac{d}{dx}(x^{m/n})=\frac{m}{n}x^{(m/n)-1}[/latex].
Derivative of the natural exponential function
[latex]\frac{d}{dx}(e^{g(x)})=e^{g(x)} g^{\prime}(x)[/latex]
Derivative of the natural logarithmic function
[latex]\frac{d}{dx}(\ln (g(x)))=\dfrac{1}{g(x)} g^{\prime}(x)[/latex]
Derivative of the general exponential function
[latex]\frac{d}{dx}(b^{g(x)})=b^{g(x)} g^{\prime}(x) \ln b[/latex]
Derivative of the general logarithmic function
[latex]\frac{d}{dx}(\log_b (g(x)))=\dfrac{g^{\prime}(x)}{g(x) \ln b}[/latex]

logarithmic differentiation: is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly