# Exponential Functions Exponential functions are functions whose input is the exponent of a constant real term. They can be expressed using the following form $ f(x) = c^x $ where $c$ is a real constant term and the domain of $f$ is limited to real numbers. Exponential equations can be solved by applying the [[Exponent Rules]] and [[Logarithm Rules]]. ## Exponential Function Derivative $\frac d {dx} \bigg[b^x \bigg] = b^x\ln(b)$ ^x4xm54 ### Proof If follow [[Exponent Rules#^k5kjan|the exponent change of base formula]] we can express $b^x$ with base $e$: $b^x = e ^ {\ln(b) x}$ We can then take derivative of both sides and apply [[The Derivative Chain Rule]] to the right side: $\frac d {dx} \bigg[b^x \bigg] = e^{\ln(b)x} \ln(b)$ Now we can reverse then change of base from the first step, which gives us the derivative of $b^x$: $\frac d {dx}\bigg[b^x\bigg] = b^x\ln(b)$ ## Exponential Function Antiderivative $\int e^{ax}dx = \frac 1 a e^{ax} + C$ $\int a^{nx} dx = \frac 1 {n \ln a} a^{nx} + C$ ^8n8b22 ### Proof $ \begin{align} \int e^{ax} \, dx &\; \text {(substitute } u=ax \; du=a \, dx) \\ \int e^u \frac 1 a \, du &= \frac 1 a \int e^u \, du = \frac 1 a e^u + C \end{align} $ $\int a^{nx}dx = \int e^{n\ln a \cdot x} dx = \frac 1 {n\ln a}e^{n\ln a \cdot x} + C = \frac 1 {n \ln a}a^{nx} + C$ ## The Exponential Function Is Not an Exponential Function Exponential functions represent a categorization of functions in algebra, while [[The Exponential Function]] represents a different concept. It is unfortunate that the naming of the two terms collide, especially because (in the strict sense) the exponential function is not an exponential function. While exponential functions are defined in the form $c^x$, the exponential function is defined as ![[The Exponential Function#^fo2wdr]] While it is true that the values of $\exp$ for real inputs equal that of $e^x$, it is not fair to say that the two functions are the same because their domains and codomains are different. The domain of $e^x$ are real numbers (exponentiation using complex numbers is non-sensical), while the domain of $\exp$ is the whole complex set. What complicates matters further is the fact that $e^x$ is often used as a shorthand for $\exp$. ![[The Exponential Function#^vm4wyz]]