Derivatives of Logarithmic Functions

Core Idea

The derivative of the natural logarithm is d/dx[ln(x)] = 1/x. For a general base, d/dx[log_b(x)] = 1/(x * ln(b)). With the chain rule, d/dx[ln(g(x))] = g'(x)/g(x). Logarithmic differentiation is a technique where you take ln of both sides before differentiating, which simplifies products, quotients, and variable exponents. The result d/dx[ln(x)] = 1/x is also why the integral of 1/x is ln|x| + C.

How It's Best Learned

Derive d/dx[ln(x)] using inverse function differentiation: if y = ln(x), then e^y = x, differentiate implicitly. Practice chain rule applications: ln(x² + 1), ln(sin(x)). Introduce logarithmic differentiation for expressions like x^x or (x² + 1)^(sin(x)).

Common Misconceptions

• Forgetting the chain rule: d/dx[ln(2x)] = 1/x (the 2 cancels), but d/dx[ln(x²)] = 2/x.
• Confusing d/dx[ln(x)] = 1/x with d/dx[e^x] = e^x.
• Not recognizing when logarithmic differentiation simplifies a problem.

Explainer

You know that d/dx[eˣ] = eˣ — the exponential function is its own derivative. The natural logarithm ln(x) is the inverse function of eˣ, and that relationship lets you derive its derivative without memorizing anything new. Set y = ln(x), so eʸ = x. Differentiate both sides with respect to x (using the chain rule on the left): eʸ · (dy/dx) = 1. Solve for dy/dx: dy/dx = 1/eʸ = 1/x. That's it — d/dx[ln(x)] = 1/x. This result is geometrically sensible: the slope of y = ln(x) is large and positive near x = 0 (the curve rises steeply) and approaches 0 as x grows (the curve flattens). The function 1/x captures exactly that behavior.

The chain rule upgrades this to composite functions. If u = g(x) is a differentiable function, then d/dx[ln(g(x))] = g'(x)/g(x). The derivative of the argument appears in the numerator; the argument itself stays in the denominator. Examples: d/dx[ln(x² + 1)] = 2x/(x² + 1), and d/dx[ln(sin(x))] = cos(x)/sin(x) = cot(x). For general bases, d/dx[log_b(x)] = 1/(x · ln(b)), which follows from rewriting log_b(x) = ln(x)/ln(b) and differentiating. The ln(b) factor in the denominator explains why natural logarithms (base e) are "natural" for calculus — the ln(e) = 1 makes the formula simplest.

Logarithmic differentiation is the technique that makes d/dx[ln(x)] = 1/x genuinely powerful beyond its own derivative. When you need to differentiate a complicated product, quotient, or function with a variable in the exponent — like f(x) = xˣ or f(x) = (x²+1)^{sin x} — take the natural log of both sides first: ln(f) = sin(x)·ln(x²+1). Now differentiate both sides (using the chain rule on the left: f'/f) and solve for f'. This technique converts exponential, product, and power relationships into additions and subtractions, which are far easier to handle. The result f'/f is called the logarithmic derivative, and it appears throughout calculus and beyond — in probability, physics, and the study of functions with multiplicative structure.