Question

Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is ______.

Options

• 0

• 1

• `1/sqrt(2)`

• `sqrt(2)`

Answer 1

Derivative of log (sec θ + tan θ) with respect to sec θ at θ = `π/4` is 1.

Explanation:

Let u = log (sec θ + tan θ) and v = sec θ

After differentiating on both sides w.r.t. θ, we get

`(du)/(dθ) = 1/((secθ + tanθ))(secθ tanθ + sec^2θ)`

`(dv)/(dθ)` = sec θ tan θ

`(du)/(dv) = ((du)/(dθ))/((dv)/(dθ))`

= `(secθ(tanθ + secθ))/((secθ + tanθ) xx secθtanθ)`

= cot θ

Hence, `((du)/(dv))_((θ = π/4)) = cot  π/4` = 1