Derivative of ln(x)
The natural log has the most surprising derivative in first-year calculus: a logarithm goes in, and a simple reciprocal comes out.
Step-by-step solution
You are differentiating f(x) = ln(x), the logarithm base e, defined for x > 0.
If y = ln(x) then ey = x. Differentiating both sides implicitly: ey · y′ = 1.
y′ = 1/ey = 1/x, since ey is just x.
Why it works
This result is the bridge between two worlds: it's the reason the area under the curve 1/t from 1 to x equals ln(x). In many textbooks that integral is literally the definition of the natural log, and the derivative formula is then true by construction. Either way you approach it, ln and 1/x are permanently linked.
The slope 1/x also explains the shape of the log curve: enormous slope near zero (the graph shoots down toward −∞), slope exactly 1 at x = 1, and an ever-flattening climb after that. The log never stops growing, but its growth rate decays to nothing — which is why logarithmic growth is the standard example of "technically unbounded, practically slow."
Domain and tangent-line example
For real-valued calculus, ln(x) is defined only for x > 0. Its derivative 1/x applies on that same domain. Although 1/x also exists at negative inputs, that does not extend the domain of ln(x). At x = 0, neither the function nor its derivative is defined.
At x = 1, the graph passes through (1, 0) because ln(1) = 0, and its slope is 1/1 = 1. The tangent-line formula y − f(a) = f′(a)(x − a) therefore gives y = x − 1. At x = 2 the slope is 1/2, illustrating how the curve becomes less steep as x increases.
For a composite natural logarithm, use the chain rule: d/dx[ln(g(x))] = g′(x)/g(x) wherever g is differentiable and g(x) > 0. For ln(2x), this is 2/(2x) = 1/x with x > 0. For ln(x²), it is 2x/x² = 2/x with x ≠ 0, since x² is positive on both sides of zero.
Common mistakes
- Writing ln(x)/x or 1/ln(x) — the answer is a plain reciprocal of x, with no logarithm left in it.
- Forgetting the chain rule on composite logs: d/dx[ln(3x² + 1)] = 6x/(3x² + 1), not 1/(3x² + 1).
- Using this formula for log base 10. Only the natural log differentiates to exactly 1/x; other bases pick up a 1/ln(base) factor.
Practice problems
Differentiate 7 ln(x)
Answer: 7/x.
Differentiate ln(x) + x²
Answer: 1/x + 2x.
Find the slope of ln(x) at x = e
Answer: 1/e ≈ 0.368.