Derivative of eˣ
Lesson notes
Overview
The only function (up to scaling) that is its own derivative. This topic is rated intermediate and is number 63 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.
What you are seeing
The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
The function eˣ is special because its slope at every point equals its height: d/dx eˣ = eˣ. Up to a constant multiple it is the only function with this property. That is why exponential growth models a quantity whose rate of change is proportional to its size. Combined with the chain rule, it also gives the derivative of e^(kx) as k·e^(kx), which is the core of solving many differential equations.
Reading the relationship
The expression "d/dx eˣ = eˣ" is a compact summary of the idea. It uses the symbols d, x, e; name what each one stands for before you start. Notice that it contains a ratio, so increasing the denominator makes the result smaller, and an exponential, which changes by the same factor for every equal step. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.
Try this
Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.
Common pitfalls
A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.
How it connects
It builds on “Chain Rule”, the topic before it in Mathematics. Once you are comfortable, continue with “Limits”.
Why it matters
Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.
Where you might use it
- Revising “Derivative of eˣ” before a class, exam or certification
- Checking working on homework and exam questions by watching the graph respond
- Modelling growth, change and uncertainty in finance, data science and statistics
- Building the reasoning behind the algorithms and engineering tools people use every day
Frequently asked questions
What is Derivative of eˣ?
The only function (up to scaling) that is its own derivative. It is an intermediate topic in Mathematics.
How do I use the Derivative of eˣ simulation?
The panel at the top shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression d/dx eˣ = eˣ packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Builds on the basics with relationships, methods and trade-offs. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Chain Rule”, the previous topic in Mathematics. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.