Derivative of ln(1+x)
2026-02-28 06:06 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-ln(1+x)

We can derive the derivative of ln(1+x) using proofs.

To demonstrate this, we utilize properties of logarithms and rules of differentiation. Some methods to prove this include:

By First Principle

Using Chain Rule

We will demonstrate how to derive the derivative of ln(1+x) using these methods:

By First Principle

The derivative of ln(1+x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of ln(1+x) using the first principle, consider f(x) = ln(1+x).

Its derivative can be expressed as the following limit: f'(x) = limₕ→₀ [f(x+h) - f(x)] / h … (1)

Given f(x) = ln(1+x), we write f(x+h) = ln(1+(x+h)).

Substituting these into equation (1), f'(x) = limₕ→₀ [ln(1+x+h) - ln(1+x)] / h = limₕ→₀ ln[(1+x+h)/(1+x)] / h Using the property ln(a/b) = ln(a) - ln(b), = limₕ→₀ ln[1 + h/(1+x)] / h

Using the limit property, limₕ→₀ ln(1+y)/y = 1 as y→0, f'(x) = 1/(1+x)

Hence, proved.

Using Chain Rule

To demonstrate the differentiation of ln(1+x) using the chain rule, Consider u = 1+x.

Then, ln(1+x) = ln(u). Using the chain rule formula: d/dx [ln(u)] = 1/u · du/dx

Since u = 1+x, du/dx = 1.

Substitute into the chain rule formula: d/dx (ln(1+x)) = 1/(1+x) · 1 = 1/(1+x)

Thus, the derivative is 1/(1+x).