Differentiating Exponentials & Logs with Other Bases
Other bases extend the rules to \(a^{x}\) and \(\log_a x\), using \(\dfrac{d}{dx}a^{x}=a^{x}\ln a\) and \(\dfrac{d}{dx}\log_a x=\dfrac{1}{x\ln a}\).
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
Exponentials and logs to a base other than \(e\) pick up a factor of \(\ln a\). This Year 12 Mathematics Advanced topic (MAV-12-04) differentiates \(a^{x}\) and \(\log_a x\).
\(\dfrac{d}{dx}a^{x}=a^{x}\ln a\) and \(\dfrac{d}{dx}\log_a x=\dfrac{1}{x\ln a}\). With the chain rule, \(\dfrac{d}{dx}a^{f(x)}=f'(x)a^{f(x)}\ln a\).
When \(a=e\), \(\ln e=1\), so these collapse to \(e^{x}\) and \(\dfrac1x\).
Method
- Identify the base \(a\).
- For \(a^{x}\) multiply by \(\ln a\); for \(\log_a x\) divide by \(\ln a\).
- Apply the chain rule for a function in the exponent.
| \(\dfrac{dy}{dx}\) | \(=\) | \(2^{x}\ln2\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(\dfrac{1}{x\ln10}\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(2\cdot3^{2x}\ln3\) |
| \(\dfrac{dy}{dx}\) | \(=\) | \(5^{x}\ln5\) |
| \(\text{at }x=0\) | \(=\) | \(\ln5\) |
Common pitfalls
Frequently asked questions
What is the derivative of a to the x?
It is a to the x times ln a, where a is the base.
What is the derivative of 2 to the x?
It is 2 to the x times ln 2.
What is the derivative of log base 10 of x?
It is 1 divided by x times ln 10.
Why does base e not need the ln a factor?
Because ln e equals 1, so the factor of ln a is just 1 and disappears.