Differentiating Exponential Functions
Learn differentiating exponential functions for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-04). The function \(e^x\) is special because it is unchanged by differentiation, and that single fact unlocks the derivative of every exponential with base \(e\).
You will learn to differentiate \(e^x\), multiples such as \(ke^{ax}\), and composite forms like \(e^{x^2}\) using the chain rule, then use the results to find gradients — the starting point for the growth and decay models used across science and finance.
Theory
The exponential function \(e^{x}\) is its own derivative. This Year 12 Mathematics Advanced topic (MAV-12-04) differentiates \(e^{x}\), \(e^{ax}\) and \(e^{f(x)}\) using the chain rule.
\(\dfrac{d}{dx}e^{x}=e^{x}\). Using the chain rule, \(\dfrac{d}{dx}e^{f(x)}=f'(x)e^{f(x)}\): keep the exponential and multiply by the derivative of the exponent.
A constant multiple stays out the front: \(\dfrac{d}{dx}k\,e^{f(x)}=k\,f'(x)e^{f(x)}\).
Method
- Spot the exponent \(f(x)\).
- Differentiate the exponent to get \(f'(x)\).
- Write \(f'(x)e^{f(x)}\), keeping any constant multiple.
| \(\dfrac{dy}{dx}\) | \(=\) | \(3e^{3x}\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(2x\,e^{x^{2}}\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(5e^{5x}+2x\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(6e^{3x}\) |
| \(\text{at }x=0\) | \(=\) | \(6\) |
Common pitfalls
Frequently asked questions
What is the derivative of e to the x?
It is e to the x. The exponential function with base e is the only function that is its own derivative.
How do you differentiate e to the power of a function?
Use the chain rule: the derivative of e to the f of x is f prime of x times e to the f of x. Keep the exponential and multiply by the derivative of the exponent.
What is the derivative of e to the 3x?
It is 3 times e to the 3x, because the derivative of the exponent 3x is 3.
Is differentiating e to the x the same as a power rule?
No. The power rule d by dx of x to the n is n x to the n minus 1, but e to the x differentiates to itself.