Derivative Calculator

Exponential rule

An exponential grows in proportion to its own size; the constant of proportionality is \(\ln a\).

\[\frac{d}{dx}a^{x} = a^{x}\ln a\]

What it says

Exponential functions are the ones whose rate of change is proportional to their current value — populations, compound interest, radioactive decay. Differentiating \(a^{x}\) returns \(a^{x}\) again, multiplied by a fixed number.

That number is \(\ln a\). It comes out of the limit \(\lim_{h \to 0}\frac{a^{h} - 1}{h} = \ln a\), and it is what makes base \(e\) special: \(\ln e = 1\), so \(\frac{d}{dx}e^{x} = e^{x}\) exactly. \(e\) is precisely the base for which the constant is one.

Any other base can be routed through \(e\) if you prefer: \(a^{x} = e^{x\ln a}\), and the chain rule then hands back \(e^{x\ln a}\ln a = a^{x}\ln a\).

When it applies

  • The base is constant and the variable sits in the exponent.
  • \(e^{x}\) and \(\exp(x)\), the special case where the extra factor is \(1\).
  • With the chain rule when the exponent is a function: \(\left(a^{u}\right)' = a^{u}\ln a \cdot u'\).

Five worked examples

Every line is the step the calculator would show, in the order it applies them. Each graph is live: hover it to read both curves and see the tangent whose slope is the derivative, drag to pan, scroll to zoom.

Example 1
\[\frac{d}{dx}\left[e^{x}\right]\]

  1. Here \(a = e\), and \(\ln e = 1\).

    \[\frac{d}{dx}\left[e^{x}\right] = e^{x}\ln e\]

  2. The extra factor is \(1\).

    \[= e^{x}\]
Answer
\[e^{x}\]

The two curves lie exactly on top of each other. This is what "its own derivative" looks like: at every height, the steepness equals the height. Hover anywhere and the f and f′ readouts agree to the last digit.

The only function (up to a constant multiple) that is its own derivative.

Open e^x in the calculator
Example 2
\[\frac{d}{dx}\left[2^{x}\right]\]

  1. Base \(2\), so the constant is \(\ln 2\).

    \[= 2^{x}\ln 2\]
Answer
\[2^{x} \ln\left(2\right)\]

f′ has the same shape as f but sits below it, shorter by the factor \(\ln 2 \approx 0.693\) — the curves never cross.

\(\ln 2 \approx 0.693\), so \(2^{x}\) grows a little more slowly than its own height would suggest.

Open 2^x in the calculator
Example 3
\[\frac{d}{dx}\left[3^{2 x}\right]\]

  1. Exponential rule on the outside, chain rule for the exponent \(u = 2x\).

    \[= 3^{2x}\ln 3 \cdot \frac{d}{dx}\left[2x\right]\]

  2. The inner derivative is \(2\).

    \[= 2\ln(3)\,3^{2x}\]
Answer
\[2 \ln\left(3\right) 3^{2 x}\]

Both curves leave the top of the window within a couple of units of the origin, which is why the framing pulls in so tightly.

Two constants appear: \(\ln 3\) from the base, \(2\) from the chain rule.

Open 3^(2x) in the calculator
Example 4
\[\frac{d}{dx}\left[e^{-x}\right]\]

  1. Outer \(e^{u}\), inner \(u = -x\).

    \[= e^{-x}\cdot\frac{d}{dx}\left[-x\right]\]

  2. The inner derivative is \(-1\).

    \[= -e^{-x}\]
Answer
\[-e^{-x}\]

f decays toward the axis while f′ is its mirror image below it: the height of the curve is the rate at which it falls.

Exponential decay: always positive, always decreasing.

Open e^(-x) in the calculator
Example 5
\[\frac{d}{dx}\left[5 e^{3 x}\right]\]

  1. Pull the \(5\) out, then chain rule with \(u = 3x\).

    \[= 5e^{3x}\cdot 3\]

  2. Multiply the constants.

    \[= 15e^{3x}\]
Answer
\[15 e^{3 x}\]

f′ is three times f at every point — hover and compare the readouts. That constant ratio is the growth rate \(k = 3\).

The general shape \(Ae^{kx}\) differentiates to \(kAe^{kx}\) — the solution to every simple growth equation.

Open 5 e^(3x) in the calculator

Watch out for

  • Do not use the power rule here. \(2^{x}\) has a constant base, so \(\frac{d}{dx}2^{x} \neq x2^{x-1}\).
  • The \(\ln a\) factor is easy to drop for bases other than \(e\).
  • If the variable appears in both base and exponent, neither this nor the power rule applies — use logarithmic differentiation.
Logarithmic differentiationChain rulePower rule