Differentiation Of Exponential and Logarithmic Functions

The derivative of e^x is e^x

The derivative of e^x is e^x itself. This unique property makes e^x the most important exponential function. For composite functions like e^f(x), we u

For any positive constant a ≠

For any positive constant a ≠ 1, d/dx[a^x] = a^x × ln(a). This formula is derived using logarithmic differentiation. Example: d/dx[3^x] = 3^x × ln(3).

The derivative of log x (natural

The derivative of log x (natural logarithm ln x) is 1/x. For composite functions: d/dx[log f(x)] = (1/f(x)) × f'(x). Example: d/dx[log(3x+1)] = (1/(3x

A powerful technique for differentiating functions

A powerful technique for differentiating functions of the form [f(x)]^g(x) where both base and exponent are functions of x. Method: (1) Take natural l

When exponential or logarithmic functions

When exponential or logarithmic functions are composed with other functions, we must apply the chain rule systematically. For nested compositions like