Derivative of e^x is e^x

Derivative

e^x → e^xengine: e^x

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

For a general base a, the derivative of a^x is a^x ln(a). The number e is the unique base for which ln(a) = 1, so the extra factor disappears. Equivalently, e is chosen so that the slope of a^x at x = 0 is exactly 1. Every other exponential is a constant multiple of that slope, which is why e^x is the natural one to differentiate.

From the definition, (e^(x+h) - e^x)/h = e^x (e^h - 1)/h. The limit (e^h - 1)/h -> 1 is the defining property of e, so the difference quotient tends to e^x. The limit is not an approximation: it is what the number e was selected to satisfy.

A function equal to its own derivative grows at a rate proportional to its size. At x = 0 the value and the slope are both 1; at x = ln(2) both are 2; at x = 5 both are about 148.4. This is the reason e^x models unconstrained growth, and why adding a constant multiple C·e^x still solves y' = y.

From the difference quotient to the fixed point

1. Write the difference quotient (e^(x+h) - e^x)/h.
2. Factor out e^x using e^(x+h) = e^x·e^h, giving e^x (e^h - 1)/h.
3. Use the defining limit of e: (e^h - 1)/h -> 1 as h approaches 0.
4. Multiply: e^x·1 = e^x. The derivative equals the original function.
5. Contrast with a general base: d/dx a^x = a^x ln(a). Only a = e removes the factor.
6. Check at x = 0: e^0 = 1 and the slope is 1, so the tangent line is y = x + 1.
7. Check the chain rule on e^(2x): the inner derivative 2 comes down, giving 2e^(2x).

Worked example

Value and slope of e^x at x = ln(2), and the tangent at x = 0

1. At x = ln(2), e^x = 2. Because the derivative is the same function, the slope there is also 2.
2. The tangent line at that point is y - 2 = 2(x - ln(2)).
3. At x = 0 the value is 1 and the slope is 1, so the tangent line is y - 1 = x, or y = x + 1.
4. A secant from 0 to 0.001 gives (e^0.001 - 1)/0.001, about 1.0005, confirming the unit slope at the origin.

Answer: 2

Common mistakes

Related derivatives

• Exponential derivatives — The rule page lists this example and the others in the same family.
• Derivative of e^(-x) — Same family of differentiation, different function.
• Derivative of e^(2x) — Same family of differentiation, different function.
• Derivative of 2^x — Same family of differentiation, different function.
• Derivative of 10^x — Same family of differentiation, different function.

Frequently asked questions