Exponential and logarithmic derivatives
Differentiate the fundamental exponential and logarithm functions.
Exponential and logarithmic derivatives is one of 9 derivatives formulas in the calculus section of this library, and it is used at ap · university level.
Why exponential and logarithmic derivatives works
The number e is picked so that the curve y = e to the x crosses the vertical axis with slope exactly 1, and the difference quotient then rebuilds the height times that slope of 1, so the function repeats itself. Rewriting a to the x as e to the (x times ln a) shows where the extra ln a factor comes from, and the log rule follows because the logarithm undoes the exponential.
What each symbol means
$a$ is a positive constant base.
Exponential and logarithmic derivatives: when it holds
$a>0$, $a\ne1$; for real $\ln x$, $x>0$.
When it stops applying
These rules need a constant base with the variable in one place only. An expression like x to the x has the variable in the base and in the exponent at once, so neither this rule nor the power rule applies, and you have to take logarithms of both sides first.
Exponential and logarithmic derivatives: a worked example
$\frac{d}{dx}5^x=5^x\ln5$.
The mistake to avoid
What people do: Using 1 over x as the derivative of every logarithm, including log base 10.
Why it goes wrong: Only the natural logarithm gives that clean answer; log base 10 of x differentiates to 1 over (x times ln 10), which is smaller by a factor of about 2.303.
Do this instead: Convert to natural logs first by writing log base 10 of x as ln x divided by ln 10, then differentiate.
Exponential and logarithmic derivatives: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a$ is a positive constant base.
- Check the conditions before substituting. $a>0$, $a\ne1$; for real $\ln x$, $x>0$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Derivatives
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where exponential and logarithmic derivatives comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about exponential and logarithmic derivatives
Why is e to the x its own derivative?
Because e is defined so that the slope of the curve matches its own height everywhere. Any other base picks up an extra constant factor of ln a to correct the mismatch.
What is the derivative of 2 to the x?
It is 2 to the x multiplied by ln 2, about 0.693 times the function value, so this curve climbs a little more slowly than its own height.
Why does the ln rule have no absolute value bars?
The natural log only accepts positive inputs, so 1 over x is already positive. Absolute value shows up in the antiderivative of 1 over x, which also has to describe negative inputs.
Where does the ln a factor actually come from?
Rewrite a to the x as e raised to x times ln a. The chain rule then multiplies by the derivative of that exponent, and the derivative of x times ln a is the constant ln a.