The value of log ₑ 2 d/(d x)(log _(cos x) cosec x) at x=π/4 is

Solution

Maths sir · 4 steps

Switch to natural logs first. Then it is a plain quotient rule, and the ln 2 outside is there to cancel.

1. Change of base. A log with base cos x cannot be differentiated directly. Write y = log_(cos x) cosec x = (ln cosec x)/(ln cos x) = -(ln sin x)/(ln cos x), since ln cosec x = -ln sin x.
2. Quotient rule. Use d/dx ln sin x = cot x and d/dx ln cos x = -tan x (chain rule). The quotient rule then gives:

   dy/dx = -(cot x ln cos x + tan x ln sin x)/(ln cos x)²

3. Put x = π/4. Here tan x = cot x = 1 and ln sin x = ln cos x = ln 1/√2 = -1/2 ln 2. The numerator is -ln 2 and the denominator is 1/4(ln 2)², so dy/dx = (ln 2)/(1/4(ln 2)²) = 4/(ln 2).
4. So. logₑ 2 × 4/(ln 2) = 4. The answer is (d).

The idea to keep. Whenever the base of a log has x in it, rewrite it as ln(argument)/ln(base) before differentiating. Then it is only the quotient rule.

Answer from the NTA final answer key. Solution by PrepWiser.