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Differentiating the exponential function

Differentiating the exponential function

The exponential function, $e^x$ can be formally defined as follows: $$ \begin{align} e^x &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \\ &= \sum_{k=0}^\infty \frac{x^k}{k!} \end{align} $$

This gives rise to the following key property $$ \begin{align} \frac{d}{dx}e^x &= \frac{d}{dx}\left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \right)\\ &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \\ &= e^x \end{align} $$

This extends nicely, to give $$ \frac{d}{dx}e^{kx} = ke^{kx} $$ where $k$ is a constant

This can be further extended to functions of the form $a^x$, where $a$ is a constant. $$ \begin{align} \frac{d}{dx}a^x &= \frac{d}{dx}\left(e^{\ln a}\right)^x \\ &= \frac{d}{dx}e^{x\ln a} \\ &= \ln (a) e^{x\ln a} \\ &= \ln(a)a^x \end{align} $$