Derivative of ln(sec x)
2026-02-28 06:15 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-ln(secx)

We can derive the derivative of ln(sec x) using proofs. To show this, we will use the trigonometric identities along with the rules of differentiation.

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

  • By First Principle
     
  • Using Chain Rule
     
  • Using Product Rule

We will now demonstrate that the differentiation of ln(sec x) results in tan(x) using the above-mentioned methods:

By First Principle

The derivative of ln(sec 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(sec x) using the first principle, we will consider f(x) = ln(sec x). 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(sec x), we write f(x + h) = ln(sec(x + h)).

Substituting these into equation (1), f'(x) = limₕ→₀ [ln(sec(x + h)) - ln(sec x)] / h = limₕ→₀ ln([sec(x + h)/sec x]) / h = limₕ→₀ ln([cos x/cos(x + h)]) / h Using the property ln(a/b) = ln a - ln b, f'(x) = -limₕ→₀ ln(cos(x + h)/cos x) / h = -limₕ→₀ ln(1 - tan x · h + O(h²)) / h Using the approximation ln(1 + u) ≈ u for small u, f'(x) = -limₕ→₀ (-tan x · h + O(h²))/ h = tan x Hence, proved.

Using Chain Rule

To prove the differentiation of ln(sec x) using the chain rule, We use the formula: ln(sec x) = ln(1/cos x) = -ln(cos x) Consider f(x) = -ln(cos x) By the chain rule: d/dx [-ln(cos x)] = -1/cos x · d/dx (cos x) = sin x/cos x = tan x Therefore, d/dx (ln(sec x)) = tan x

Using Product Rule

We will now prove the derivative of ln(sec x) using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, ln(sec x) = ln |sec x| ln(sec x) = ln |1/cos x| = -ln(cos x) Given that, u = -1 and v = ln(cos x) Using the product rule formula: d/dx [u · v] = u' · v + u · v' u' = d/dx (-1) = 0 Here we use the chain rule: v = ln(cos x) v' = d/dx [ln(cos x)] = -sin x/cos x v' = -tan x Again, use the product rule formula: d/dx (ln(sec x)) = 0 + (-1) · (-tan x) = tan x