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Logarithmic Differentiation Calculator

Logarithmic derivative calculator. Enter f(x) and x to compute (ln f)′=f′/f, the relative rate—ideal for products, quotients and powers—with steps.

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(ln f)′ = f′ / f (logarithmic derivative)
Best for products or powers
f(x) must be positive

📖 Tutorial｜Logarithmic Differentiation

1. Definition
Logarithmic differentiation: take ln y, differentiate w.r.t. x to get y′/y=(ln y)′=f′/f; great for products, quotients and powers.
2. Symbols
SymbolMeaning
f(x)The function
f′/fLog derivative (relative rate)
ln yAfter taking logs
3. How it works
  • We numerically compute f and f′;
  • Divide to get f′/f;
  • It turns multiplication into addition.
4. Steps
  1. Enter f(x);
  2. Enter x (f must be positive);
  3. Click Calculate.
5. Example
Example: f=x²eˣ at x=1. ln f=2ln x+x, so f′/f=2/x+1=3.
6. Pitfalls
Taking logs requires f>0;
You get f′/f, remember to multiply by f for f′;
Best for products and powers, not sign-changing functions.

❓ FAQ｜Logarithmic Differentiation

Why use logarithmic differentiation?
It turns products into sums, quotients into differences and powers into multiplication.
Why do we get f′/f?
Because (ln f)′=f′/f; multiply back by f to recover f′.
Which functions suit it best?
Mixed products/quotients and powers u(x)^v(x) such as x^x.
Must f be positive?
ln requires f>0; if f may be negative, take |f| first.
What is the relative rate?
f′/f is the change per unit of function value, used for growth rates.