Derivative of Exponential Function

Theorem
Let $\exp x$ be the exponential function.

Then:
 * $D_x \left({\exp x}\right) = \exp x$

Corollary 1
Let $c \in \R$.

Then:
 * $D_x \left({\exp \left({c x}\right)}\right) = c \exp \left({c x}\right)$

Corollary 2
Let $a \in \R: a > 0$.

Let $a^x$ be $a$ to the power of $x$.

Then:
 * $D_x \left({a^x}\right) = a^x \ln a$

Proof
We have:

From one of the definitions of the exponential function,

The right summand clearly converges to zero as $h$ gets arbitrarily small, and so $\displaystyle \lim_{h \to \infty}\frac{\exp h - 1}{h} = 1$.

From Combination Theorem for Functions:
 * $\displaystyle \lim_{h \to 0} \frac {\exp x \left({\exp h - 1}\right)} {h} = \exp x \left({\lim_{x \to 0} \frac {\exp h - 1} {h}}\right)$

The result follows.

Alternative proof
We use the definition of the exponential function as the inverse of the natural logarithm function.

Let $y = \exp x$.

Then from Derivative of an Inverse Function, we have:


 * $\displaystyle D_x \exp x = \frac 1 {D_x \ln x} = \frac 1 {1/y} = y = \exp x$

Proof of Corollary 1
Follows directly from the Derivative of Function of Constant Multiple:


 * $\displaystyle D_x \left({\exp \left({c x}\right)}\right) = c D_{c x} \left({\exp \left({c x}\right)}\right) = c \exp \left({c x}\right)$

Proof of Corollary 2
From the definition, $a^x = e^{x \ln a}$.

Thus from Corollary 1:
 * $D_x \left({a^x}\right) = D_x \left({e^{x \ln a}}\right) = \ln a e^{x \ln a} = a^x \ln a$