Logarithmic Differentiation: Step-by-Step Guide
Logarithmic differentiation simplifies derivatives of complex products, quotients, and especially functions with variable bases AND variable exponents (like x^x). Take the natural log of both sides, use log properties to break the expression apart, differentiate implicitly, then solve for y'.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Take the natural log of both sides. Set y = f(x) and write ln(y) = ln(f(x)). This converts products to sums, quotients to differences, and exponents to multipliers.\[\ln y = \ln(f(x))\]
- Apply log properties to expand. Use ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), and ln(a^n) = n*ln(a) to simplify the right-hand side into a sum of simpler terms.\[\ln(a^n b^m / c^k) = n\ln a + m\ln b - k\ln c\]
- Differentiate both sides implicitly. The left side d/dx[ln y] = (1/y)(dy/dx). Differentiate the right side term-by-term.\[\frac{1}{y} \cdot \frac{dy}{dx} = \frac{d}{dx}[\text{RHS}]\]
- Solve for dy/dx and substitute. Multiply both sides by y, then replace y with the original f(x) to express dy/dx purely in x.\[\frac{dy}{dx} = y \cdot \frac{d}{dx}[\text{RHS}] = f(x) \cdot \frac{d}{dx}[\text{RHS}]\]
Worked example
Find dy/dx if y = x^x.
Step 1: Take ln of both sides: ln(y) = ln(x^x).
Step 2: Apply the log power property: ln(y) = x * ln(x).
Step 3: Differentiate both sides. Left: (1/y)(dy/dx). Right: use the product rule on x*ln(x): d/dx[x*ln(x)] = 1*ln(x) + x*(1/x) = ln(x) + 1.
Step 4: So (1/y)(dy/dx) = ln(x) + 1.
Step 5: Multiply both sides by y: dy/dx = y(ln(x) + 1).
Step 6: Substitute y = x^x: dy/dx = x^x (ln(x) + 1).
Verification: At x = 1, dy/dx = 1*(0 + 1) = 1. Numerical: (1.001^1.001 - 1)/0.001 approximately 0.999, confirming about 1.
Tips and common mistakes
- Use logarithmic differentiation whenever the function has a variable base AND a variable exponent.
- It is also handy for products/quotients of many factors — log turns them into easy sums.
- After taking ln, expand fully using log rules BEFORE differentiating.
- Don't forget the (1/y) factor on the left when you differentiate ln(y) implicitly.
- Always substitute y = original f(x) back at the end so your answer is in terms of x.
- If y could be negative, technically write ln|y|; the resulting dy/dx formula is the same.
- For x^x specifically, the answer x^x(ln(x) + 1) is worth memorizing.
- Compare logarithmic differentiation with the power rule and exponential rule — using the wrong one on x^x gives wrong results.