Derivative of e^x using First Principle of Derivatives

Using the first principle of derivatives, we will show that the derivative of

e^x

e^x

.

Proof. Let

f(x) = e^x

. We will be using the first principle derivative:

\begin{align*} f'(x) &= \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} \\ &= \lim_{h \rightarrow 0} \frac{e^{x+h} - e^x}{h} \\ &= \lim_{h \rightarrow 0} \frac{e^x(e^h - 1)}{h} \\ &= e^x \cdot \lim_{h \rightarrow 0} \frac{e^h - 1}{h}. \end{align*}

We have seen here that

\lim_{h \rightarrow 0} \frac{(e^h - 1)}{h} = 1

. So we have that

\begin{align*} f'(x) = e^x \cdot \lim_{h \rightarrow 0} \frac{e^h - 1}{h} = e^x, \end{align*}

which proves that the derivative of

e^x

e^x