Differentiation Of Exponential and Logarithmic Functions

The derivative of \( e^x \)

The derivative of \( e^x \) with respect to \( x \) is \( e^x \). This means the rate of change of the function at any point is equal to the value of

For any positive constant \(

For any positive constant \( a \), the derivative of \( a^x \) is \( a^x \log a \). This is derived by expressing \( a^x \) as \( e^{x \log a} \) and

The derivative of \( \log x

The derivative of \( \log x \) (natural logarithm) is \( \frac{1}{x} \). For \( \log(ax + b) \), the derivative is \( \frac{a}{ax + b} \), again using

When differentiating functions of the form

When differentiating functions of the form \( y = [f(x)]^{g(x)} \), such as \( x^x \) or \( (\sin x)^{\log x} \), we take the natural logarithm on bot

The second derivative is obtained by

The second derivative is obtained by differentiating the first derivative. For example, if \( y = e^x \), then \( \frac{dy}{dx} = e^x \) and \( \frac{