Logarithmic differentiation is a method for finding derivatives by first taking the natural logarithm of both sides of an equation. It is especially useful when a function contains products, quotients, roots, or variables in both the base and exponent. The method turns complicated multiplication and division into addition and subtraction, making the derivative easier to manage.
It matters because many functions that look difficult at first become straightforward after applying log rules.
The main idea is to write y = f(x), take ln y = ln(f(x)), simplify using logarithm properties, and then differentiate implicitly. Because d/dx[ln y] = y'/y, you multiply by y at the end to solve for y'. For example, y = x^x sqrt(x^2 + 1) / sin x becomes ln y = x ln x + (1/2)ln(x^2 + 1) - ln(sin x).
After differentiating, the final derivative is y times the simplified derivative of ln y.
Key Facts
- If y = f(x), then logarithmic differentiation starts with ln y = ln(f(x)).
- d/dx[ln y] = y'/y when y is a function of x.
- ln(ab) = ln a + ln b and ln(a/b) = ln a - ln b.
- ln(a^r) = r ln a, which is useful for roots, powers, and variable exponents.
- For y = x^x, ln y = x ln x, so y' = x^x(ln x + 1).
- For y = x^x sqrt(x^2 + 1) / sin x, y' = y[ln x + 1 + x/(x^2 + 1) - cot x].
Vocabulary
- Logarithmic differentiation
- A differentiation method that takes the natural logarithm of both sides before differentiating.
- Natural logarithm
- The logarithm with base e, written ln x, where e is approximately 2.718.
- Implicit differentiation
- A method of differentiating equations where y is treated as a function of x.
- Variable exponent
- An exponent that contains the variable, such as x in x^x or sin x in x^(sin x).
- Logarithm properties
- Rules that rewrite logs of products, quotients, and powers into simpler expressions.
Common Mistakes to Avoid
- Forgetting to multiply by y at the end, which is wrong because differentiating ln y gives y'/y, not y'.
- Writing ln(a + b) = ln a + ln b, which is wrong because logarithm sum rules apply to products, not addition inside the logarithm.
- Differentiating x^x as x x^(x - 1), which is wrong because the power rule only applies when the exponent is constant.
- Ignoring domain restrictions, which is wrong because taking ln y, ln x, or ln(sin x) requires the logged quantity to be positive on the interval being used.
Practice Questions
- 1 Use logarithmic differentiation to find dy/dx for y = x^3 sqrt(x^2 + 4) / (x + 1), assuming x > 0.
- 2 Use logarithmic differentiation to find dy/dx for y = x^(sin x), assuming x > 0.
- 3 Explain why logarithmic differentiation is a good strategy for y = (x^2 + 1)^5 (3x - 2)^4 / sqrt(x^2 - 9), and identify which log properties would be used.