$\frac{d}{d x}\left(e^{x}\right)=e^{x}$ #### Proof Suppose $f(x)=a^x$ $\begin{aligned} \frac{d}{d x}\left(a^{x}\right) &=\lim _{h \rightarrow 0} \frac{a^{x+h}-a^{x}}{h} \\ &=\lim _{h \rightarrow 0} \frac{a^{x} \cdot a^{h}-a^{x}}{h} \\ &=\lim _{h \rightarrow 0} a^{x} \cdot \frac{a^{h}-1}{h} \\ &=a^{x} \cdot \lim _{h \rightarrow 0} \frac{a^{h}-1}{h} \\ &=\left(\lim _{h \rightarrow 0} \frac{a^{h}-1}{h}\right) \cdot a^{x}\\ &=(\ln a) \cdot a^x \end{aligned}$ [[Table of Common limits]] see [[𝑒 - Euler's Number]]