Derivative of ln(x+2)
2026-02-28 06:07 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-ln(x+2)

We can derive the derivative of ln(x+2) using proofs.

To show this, we will use the rules of differentiation.

There are several methods we use to prove this, such as:

Using Chain Rule

By First Principle

We will now demonstrate that the differentiation of ln(x+2) results in 1/(x+2) using the above-mentioned methods:

Using Chain Rule

To prove the differentiation of ln(x+2) using the chain rule, Let u = x+2, so ln(x+2) = ln(u).

By chain rule: d/dx [ln(u)] = 1/u * du/dx.

Since u = x+2, du/dx = 1.

Thus, d/dx [ln(x+2)] = 1/(x+2).

By First Principle

The derivative of ln(x+2) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of ln(x+2) using the first principle, we will consider f(x) = ln(x+2).

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

Given that f(x) = ln(x+2), we write f(x + h) = ln(x + 2 + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [ln(x + 2 + h) - ln(x+2)] / h

Using the logarithm property ln(a) - ln(b) = ln(a/b), f'(x) = limₕ→₀ ln[(x + 2 + h)/(x + 2)] / h

Using the well-known limit property, we find: f'(x) = 1/(x+2)

Hence, proved.