Derivative of Exponential Function/Proof 2

Theorem

Let $\exp$ be the exponential function.

Then:

$\map {\dfrac \d {\d x} } {\exp x} = \exp x$

Proof

We use the fact that the exponential function is the inverse of the natural logarithm function:

$y = e^x \iff x = \ln y$

\(\ds \dfrac {\d x} {\d y}\)

\(=\)

\(\ds \dfrac 1 y\)

Derivative of Natural Logarithm Function

\(\ds \leadsto \ \ \)

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds \dfrac 1 {1 / y}\)

Derivative of Inverse Function

\(\ds \)

\(=\)

\(\ds y\)

\(\ds \)

\(=\)

\(\ds e^x\)

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.4$