Key takeaways
- d/dx ln(x) = 1/x, and d/dx ln(ax) = 1/x for any positive a.
- The a disappears because ln(ax) = ln(a) + ln(x), and ln(a) is a constant.
- With a +b inside, it changes: d/dx ln(ax + b) = a/(ax + b).
- x must be positive, because ln of zero or a negative number is not defined.
In this guide
What is the derivative of ln?
ln(x) is the natural logarithm: the power you raise e (about 2.718) to in order to get x. Its graph rises quickly at first and then flattens out.
A derivative is the slope of the graph at a point. For ln(x), the slope is 1/x: steep for small x (at x = 0.5 the slope is 2) and gentle for large x (at x = 10 it is 0.1).
Why does the derivative of ln matter?
The slope 1/x says something useful: the bigger x gets, the slower ln(x) grows. That matches how people feel many things. Going from 1 to 2 kg feels like a big jump; going from 100 to 101 kg hardly feels like anything.
Changing a in ln(ax) just slides the whole graph up or down. The shape doesn’t change, so the slope doesn’t change either.
The formula
| Symbol | What it means | Example |
|---|---|---|
| ln | The natural logarithm (log base e) | ln(20) ≈ 3.00 |
| a | A positive constant multiplying x | 5 |
| x | The point where you want the slope. Must be positive | 4 |
| 1/x | The slope, which the calculator works out | 1/4 = 0.25 |
Why the a disappears
Log rules let you split a product: ln(ax) = ln(a) + ln(x). The ln(a) part is just a fixed number, and fixed numbers have derivative 0. Only ln(x) is left, and its derivative is 1/x.
The chain rule agrees: the derivative of ln(u) is 1/u times the derivative of u. With u = ax, that is (1/(ax)) × a = 1/x.
How to differentiate ln step by step
- 1Check what is inside. Is it just a number times x, like 5x? Or is there a +b, like 3x + 1?
- 2For ln(ax), write 1/x. The a cancels out.
- 3For ln(ax + b), use the chain rule. The answer is a/(ax + b).
- 4Put in x for a slope at a point. Make sure x is positive.
Now try it yourself. Change a in the calculator below and watch the answer stay the same.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using 1/x when there’s a +b inside.
ln(ax + b)has derivativea/(ax + b), not 1/x. - Plugging in x ≤ 0. ln is only defined for positive inputs, so the calculator needs a > 0 and x > 0.
- Mixing up ln and log.
d/dx log₁₀(x) = 1/(x ln 10) ≈ 0.4343/x. Use the logarithm calculator to check log values.
Tips and tricks
- Numbers in front stay:
d/dx 5ln(x) = 5/x. - Negative x: for ln|x|, the derivative is still 1/x for any x that isn’t 0.
- Reverse: because of this rule,
∫(1/x) dx = ln|x| + C. And ln is the reverse of eˣ, covered in the derivative of eᵃˣ guide.
Where you’ll use it in real life
- Sound: decibels use logarithms, so the slope tells you how loudness changes with power.
- Chemistry: pH is a logarithm of concentration.
- Economics: ln is used to measure percentage growth, and its derivative 1/x turns changes into percentages.
- Earthquakes: magnitude scales are logarithmic.
Quick summary
The derivative of ln(ax) is 1/x for any positive a, because ln(a) is just a constant that disappears. With a +b inside, use a/(ax + b) instead. Use the calculator above to find the slope at any positive x, with every step shown.
Worked example
Your textbook asks for the slope of y = ln(5x) at x = 4, and you want to confirm that the 5 makes no difference. Here a = 5 and x = 4.
- 01d/dx[ln(5x)] = d/dx[ln(5) + ln(x)] = 1/x
- 02The constant ln(5) disappears after differentiation
- 03At x = 4: 1/4 = 0.25
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the derivative of ln 2x?
It is 1/x. Since ln(2x) = ln 2 + ln x and ln 2 is a constant, only the derivative of ln x, which is 1/x, remains.
What is the derivative of ln(x²)?
It is 2/x. Rewrite ln(x²) as 2ln(x), then differentiate to get 2 × 1/x.
What is the derivative of ln(ax + b)?
It is a/(ax + b). The +b means you cannot split the log, so use the chain rule: 1/(ax + b) times a.
Why must x be positive?
ln(x) is only defined for x > 0. For negative x, use ln|x|, whose derivative is also 1/x.
