Derivative of logn
2026-02-28 11:45 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-logn

We can derive the derivative of logn using proofs. To show this, we will use the logarithmic identities along with the rules of differentiation.

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

  1. By First Principle
  2. Using Chain Rule
  3. Using Power Rule

We will now demonstrate that the differentiation of logn results in 1/n using the above-mentioned methods:

By First Principle

The derivative of logn can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of logn using the first principle, we will consider f(n) = logn.

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

Given that f(n) = logn, we write f(n + h) = log(n + h).

Substituting these into equation (1), f'(n) = limₕ→₀ [log(n + h) - logn] / h = limₕ→₀ [ln((n + h)/n)] / h = limₕ→₀ [ln(1 + h/n)] / h Using the formula ln(1 + u) ≈ u for small u, f'(n) = limₕ→₀ (h/n) / h = 1/n

Hence, proved.

Using Chain Rule

To prove the differentiation of logn using the chain rule, We use the formula: logn = ln(n) Consider g(n) = ln(n)

The derivative is 1/n Thus: d/dn (logn) = 1/n

Using Power Rule

We will now prove the derivative of logn using the power rule. The step-by-step process is demonstrated below:

Here, we use the formula, logn = ln(n)

Given that, u = n

Using the power rule formula: d/dn (u^1) = 1 * u^(1-1) 1/n Thus: d/dn (logn) = 1/n