Mathwords logoMathwords

Logarithmic Differentiation — Definition & Examples

Logarithmic Differentiation

A method for finding the derivative of functions such as y = xsin x and y = ((x−7)^6 · √(2x+3)) / (4th root of (x+1)).

 

Example showing logarithmic differentiation of y=x^(sin x), using ln, derivatives, and solving for y' = x^(sin x)[(cos x)(ln...

Example 2 of logarithmic differentiation: finding y' for y=(x-7)^5·√(2x+3)/sixth-root(x+1) using ln, derivative rules, and...

 

See also

Derivative rules

Worked Example

Problem: Find the derivative of y = x^x for x > 0.
Step 1: Take the natural logarithm of both sides.
ln⁡y=ln⁡(xx)=xln⁡x\ln y = \ln(x^x) = x \ln x
Step 2: Differentiate both sides with respect to x. On the left, use implicit differentiation. On the right, use the product rule.
1y dydx=1⋅ln⁡x+x⋅1x=ln⁡x+1\frac{1}{y}\,\frac{dy}{dx} = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1
Step 3: Solve for dy/dx by multiplying both sides by y, then substitute the original expression for y.
dydx=y(ln⁡x+1)=xx(ln⁡x+1)\frac{dy}{dx} = y(\ln x + 1) = x^x(\ln x + 1)
Answer: dydx=xx(ln⁡x+1)\frac{dy}{dx} = x^x(\ln x + 1)

Why It Matters

Standard derivative rules (power rule, exponential rule) fail when both the base and the exponent contain the variable, as in xxx^x. Logarithmic differentiation converts such expressions into forms where the product rule and chain rule apply. It also simplifies derivatives of long products or quotients by turning multiplication into addition via logarithm properties.

Common Mistakes

Mistake: Forgetting to multiply by y (or the original function) in the final step.
Correction: When you differentiate ln y, you get (1/y)(dy/dx). You must multiply both sides by y at the end to isolate dy/dx, then replace y with the original expression.

Related Terms