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Derivative of Natural Logarithm

Complete guide with formula, proof, examples, and graph.

Quick Answer

ddxln⁡(x)=1x\frac{d}{dx}\ln(x) = \frac{1}{x}

The derivative of \ln(x) is:

Proof / Derivation

Step-by-step derivation of the derivative formula.

Start with the definition of the natural logarithm as the inverse of the exponential function.

Lety=ln⁡(x),thenx=eyLet y = \ln(x), then x = e^y

Differentiate both sides with respect to x using implicit differentiation.

Differentiateimplicitly:ddx[x]=ddx[ey]Differentiate implicitly: \frac{d}{dx}[x] = \frac{d}{dx}[e^y]

Apply the chain rule to the right side: d/dx[e^y] = e^y · dy/dx.

1=ey⋅dydx1 = e^y \cdot \frac{dy}{dx}

Solve for dy/dx and substitute back e^y = x.

dydx=1ey=1x\frac{dy}{dx} = \frac{1}{e^y} = \frac{1}{x}

Graph

Visualization of Natural Logarithm and its derivative.

f(x) = \ln(x)

f(x)=ln⁡(x)f(x) = \ln(x)

f'(x) = \frac{1}{x}

f′(x)=1xf'(x) = \frac{1}{x}
Domain: (0,+∞)(0, +\infty)Range: (−∞,+∞)(-\infty, +\infty)

Worked Examples

Step-by-step solutions using the chain rule and other techniques.

Find: ddxln⁡(5x+2)\frac{d}{dx}\ln(5x+2)

Solution: 55x+2\frac{5}{5x+2}

1.Chainrule:u=5x+2,u′=5Chain rule: u = 5x+2, u' = 5
2.ddxln⁡(5x+2)=15x+2⋅5=55x+2\frac{d}{dx}\ln(5x+2) = \frac{1}{5x+2} \cdot 5 = \frac{5}{5x+2}

Find: ddxln⁡(x2+1)\frac{d}{dx}\ln(x^2+1)

Solution: 2xx2+1\frac{2x}{x^2+1}

1.u=x2+1,u′=2xu = x^2+1, u' = 2x
2.=1x2+1⋅2x= \frac{1}{x^2+1} \cdot 2x

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Frequently Asked Questions

What is the derivative of Natural Logarithm?

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The derivative of ln⁡(x)\ln(x) is 1x\frac{1}{x}. This is one of the fundamental derivatives in calculus that you should memorize.

How do you prove the derivative of Natural Logarithm?

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The proof uses the limit definition of the derivative. See the Proof section above for the complete step-by-step derivation.

Is the derivative of Natural Logarithm always the same?

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Yes, the derivative formula for ln⁡(x)\ln(x) is constant — it does not depend on x. However, when composed with inner functions (e.g., ln⁡(x)\ln(x) of u(x)), the chain rule applies.

Where is the derivative of Natural Logarithm undefined?

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The derivative is undefined where the original function is not differentiable. Check the domain section for details.

Why is the derivative of Natural Logarithm important?

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This derivative appears frequently in physics (wave motion), engineering (signal processing), economics (oscillating models), and many other fields involving periodic or growth phenomena.