Key takeaways
- d/dx eˣ = eˣ. The function is its own derivative.
- d/dx e^(ax) = a·e^(ax): copy it, then multiply by a.
- The #1 mistake is using the power rule on eˣ.
- Other bases are different: d/dx 2ˣ = 2ˣ × ln 2.
In this guide
What is the derivative of eˣ?
The number e is about 2.71828. The function eˣ is called the exponential function. A derivative is the slope of a graph, and for most functions the slope is a different formula. The exponential is special: at every point, its slope equals its height.
At x = 0, eˣ has height 1 and slope 1. At x = 1, it has height e ≈ 2.718 and the same slope, 2.718.
Why does the derivative of eˣ matter?
The rule “slope equals height” means “the bigger it gets, the faster it grows”. That describes money earning continuous interest, bacteria doubling, and a rumour spreading. It also describes decay when a is negative: hot coffee cooling or a medicine leaving the body.
e is the only base whose exponential function is its own derivative. For any other base, like 2ˣ or 10ˣ, an extra factor appears.
The formula
| Symbol | What it means | Example |
|---|---|---|
| e | Euler’s number, about 2.71828 | 2.71828 |
| a | The growth rate: the number multiplying x in the exponent | 0.5 |
| x | The point where you want the slope (often time) | 2 |
| a·e^(ax) | The slope, which the calculator works out | 0.5 × e¹ ≈ 1.3591 |
The extra a comes from the chain rule: the exponent ax has derivative a.
How to differentiate e^(ax) step by step
- 1Find a. It is the number multiplying x in the exponent. For eˣ, a = 1.
- 2Copy e^(ax) exactly. Don’t change the exponent.
- 3Multiply by a. You now have a·e^(ax).
- 4Put in x for a value at a point. Work out ax, find e to that power, then multiply by a.
Now try it yourself. Enter a and x in the calculator below to see the rate of change with every step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting the a.
d/dx e^(2x)is2e^(2x), note^(2x). - Treating other bases the same way. For 2ˣ, the derivative is
2ˣ × ln 2 ≈ 0.6931 × 2ˣ. Only base e gives back exactly itself. - Dropping the minus in decay.
d/dx e^(−x) = −e^(−x). The slope is negative because the curve is falling.
Tips and tricks
- Numbers in front stay:
d/dx 5eˣ = 5eˣ. - Other exponents: for
e^(x²), the exponent has derivative 2x, so the answer is2x·e^(x²). - Build it from powers: the Taylor series calculator shows how eˣ is made of 1 + x + x²/2 + …, and differentiating that sum gives back the same sum.
Where you'll use it in real life
- Biology: bacteria growth, where the growth speed depends on how many there are.
- Finance: continuously compounded interest grows like e^(rt).
- Physics: radioactive decay and cooling follow e^(−kt).
- Medicine: drug levels in the blood often fall exponentially. For the amounts themselves, use the exponential growth and decay calculator.
Quick summary
The derivative of e^(ax) is a·e^(ax): copy the function and multiply by a. For plain eˣ, a = 1, so the function is its own derivative. Use the calculator above to find the rate at any point, and see the derivative of ln for the reverse function.
Worked example
A bacteria culture grows as P = e^(0.5t) thousand cells after t hours. How fast is it growing (in thousand cells per hour) at t = 2 hours? The growth rate is a = 0.5 and x = 2.
- 01d/dx[e^(0.5x)] = 0.5·e^(0.5x)
- 02At x = 2: 0.5 · e^(1)
- 03e^(1) = 2.7183
- 04= 0.5 × 2.7183 = 1.3591
These numbers are pre-filled in the calculator above.
Frequently asked questions
Why is the derivative of e^x equal to e^x?
The number e is chosen exactly so that the slope of eˣ at x = 0 is 1. Because of how exponents work, the slope is then equal to the height at every point.
What is the derivative of e^2x?
It is 2e^(2x). Keep e^(2x) as it is and multiply by 2, the derivative of the exponent.
What is the derivative of e^(−x)?
It is −e^(−x). The exponent −x has derivative −1, so you multiply by −1.
