Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions

An inverse function is a function that undoes another function.

The derivative of the exponential function is equal to the value of the function.

The logarithm of a number is the exponent by which another fixed value must be raised to produce that number.

The general form of the derivative of a logarithmic function is $\frac{d}{dx}\log_{b}(x) = \frac{1}{xln(b)}$.

Differentiation and integration of natural logarithms is based on the property $\frac{d}{dx}\ln(x) = \frac{1}{x}$.

The derivative of the exponential function $\frac{d}{dx}a^x = \ln(a)a^{x}$.

Exponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function's current value.

It is useful to know the derivatives and antiderivatives of the inverse trigonometric functions.

$\sinh$ and $\cosh$ are basic hyperbolic functions; $\sinh$ is defined as the following: $\sinh (x) = \frac{e^x - e^{-x}}{2}$.

Indeterminate forms like $\frac{0}{0}$ have no definite value; however, when a limit is indeterminate, l'Hôpital's rule can often be used to evaluate it.

Among all choices for the base $b$, particularly common values for logarithms are $e$, $2$, and $10$.