Derivative of eˣ
The exponential function is its own derivative.
Key idea
The exponential function is its own derivative.
In depth
d/dx eˣ = eˣ. The number e is defined so its curve has slope equal to its height everywhere. With a general base, d/dx aˣ = aˣ ln a.
Worked example
d/dx e^(3x) = 3e^(3x). d/dx 2ˣ = 2ˣ ln 2.
Formula
Real-world uses
- Modelling continuous growth and decay
- The rate of charge in an RC circuit
Common mistake
Applying the power rule to eˣ, getting x eˣ⁻¹.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Derivative of eˣ?
The exponential function is its own derivative. It is a topic at the advanced level in Calculus.
What is the formula for Derivative of eˣ?
The key formula is (eˣ)′ = eˣ.
Where is Derivative of eˣ used in real life?
Real-world examples: Modelling continuous growth and decay; The rate of charge in an RC circuit.
What should I learn before Derivative of eˣ?
Make sure you are comfortable with Derivatives of Trig Functions, the previous topic in Calculus.
What is a common mistake with Derivative of eˣ?
Applying the power rule to eˣ, getting x eˣ⁻¹.
How can I practise Derivative of eˣ?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Derivative of eˣ?
Move on to Derivative of ln x next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.