Logarithmic Derivative — Definition, Formula & Examples

The logarithmic derivative of a function

f(x)

is the derivative of its natural logarithm, which equals

\frac{f'(x)}{f(x)}

. It turns complicated products, quotients, and variable exponents into simpler sums and differences that are easier to differentiate.

For a differentiable function $f(x) > 0$ , the logarithmic derivative is defined as $\frac{d}{dx}[\ln f(x)] = \frac{f'(x)}{f(x)}$ . The technique of logarithmic differentiation applies $\ln$ to both sides of $y = f(x)$ , differentiates implicitly, and then solves for $y'$ .

Key Formula

\frac{d}{dx}[\ln f(x)] = \frac{f'(x)}{f(x)}

Where:

$f(x)$ = A positive differentiable function
$f'(x)$ = The ordinary derivative of f(x)

How It Works

Start by taking the natural log of both sides:

\ln y = \ln f(x)

. Use logarithm properties to expand products into sums, quotients into differences, and exponents into coefficients. Differentiate both sides with respect to

x

, remembering that

\frac{d}{dx}[\ln y] = \frac{1}{y}\frac{dy}{dx}

by the chain rule. Finally, multiply both sides by

y

to isolate

\frac{dy}{dx}

, then substitute the original expression for

y

Worked Example

Problem: Find the derivative of

y = x^x

for

x > 0

Take the natural log: Apply

\ln

to both sides and simplify the exponent.

\ln y = \ln(x^x) = x \ln x

Differentiate both sides: Use the chain rule on the left and the product rule on the right.

\frac{1}{y}\frac{dy}{dx} = (1)(\ln x) + x \cdot \frac{1}{x} = \ln x + 1

Solve for dy/dx: Multiply both sides by

y

and substitute

y = x^x

\frac{dy}{dx} = x^x(\ln x + 1)

Answer:

\dfrac{dy}{dx} = x^x(\ln x + 1)

Why It Matters

Logarithmic differentiation is essential for functions like

x^x

(\sin x)^{\cos x}

where standard derivative rules do not directly apply. It also streamlines differentiating products of many factors, a technique used in statistics when working with likelihood functions.

Common Mistakes

Mistake: Forgetting to multiply by

y

(the original function) after differentiating

\ln y

Correction: After finding

\frac{1}{y}\frac{dy}{dx} = \ldots

, you must multiply both sides by

y

and then substitute back the original expression for

y

to get the final answer.