The derivative of natural log of x is equal to 1/x. As the natural logarithm of x is denoted by lnx, so the derivative formula of natural log of x is given by

$\dfrac{d}{dx}{\ln x}=\dfrac{1}{x}$.

## Find Derivative of Natural log of x

**Answer:** The derivative of natural log of x is 1/x.

**Explanation:**

Note the natural log of x is denoted by ln(x) = log_{e}x. Let us assume that

y= ln(x) = log_{e}x. (we need to find dy/dx)

⇒ e^{y} = x.

Differentiating both sides with respect to x, we get that

e^{y} $\dfrac{dy}{dx}$ = 1

⇒ $\dfrac{dy}{dx} = \dfrac{1}{e^y}$

⇒ $\dfrac{dy}{dx}$ = e^{-y}

⇒ $\dfrac{dy}{dx}$ = e^{-ln x} = $e^{\ln x^{-1}}$

⇒ $\dfrac{dy}{dx}$ = x^{-1} = 1/x.

So the derivative of natural log of x is equal to 1/x, and this is obtained by the method of implicit differentiation.

**More Derivatives:** Derivative of a^{x} by first principle

Derivative of e^{sinx} by first principle

Derivative of root tanx by first principle

Derivative of sec2x by first principle

## Derivative of Natural log of x by First Principle

By first principle, the derivative of f(x) is given by

$\dfrac{d}{dx}$(f(x)) = lim_{h→0} $\dfrac{f(x+h)-f(x)}{h}$.

Put f(x) = ln(x).

So the derivative of ln(x), natural log of x, is given as follows:

$\dfrac{d}{dx}$( ln(x) ) = lim_{h→0} $\dfrac{\ln(x+h)-\ln(x)}{h}$

= lim_{h→0} $\dfrac{\ln(\dfrac{x+h}{x})}{h}$ by the formula: lna -lnb = ln(a/b).

= lim_{h→0} $\dfrac{\ln(1+\frac{h}{x})}{h}$

= $\dfrac{1}{x}$ lim_{h→0} $\dfrac{\ln(1+\frac{h}{x})}{h/x}$

[Let h/x=t, so t→0 as h→0]

= $\dfrac{1}{x}$ lim_{t→0} $\dfrac{\ln(1+t)}{t}$

= $\dfrac{1}{x} \times 1$

= $\dfrac{1}{x}$.

So the derivative of natural log of x by the first principle is equal to 1/x.

**Read Also:** Derivative of e^{cosx} by first principle

Derivative of $\sqrt{\sin x}$ by first principle

Derivative of $\sqrt{\cos x}$ by first principle

## FAQs

### Q1: What is the derivative of natural logarithm of x?

**Answer:** the derivative of ln x [natural log of x] is 1/x, denoted as d/dx (ln x) = 1/x (or) (ln x)’ = 1/x.