Differentiating Exponential Functions
Differentiating exponential functions uses \(\dfrac{d}{dx}e^{x}=e^{x}\) and, with the chain rule, \(\dfrac{d}{dx}e^{f(x)}=f'(x)e^{f(x)}\).
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Differential calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
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.