Question

What is the derivative of tan (sin x) + sec x
1. sec²(sin x) cos x + sin x cos x
2. sec²(sin x) cos x + sec x tan x
3. sec²(sin x) cos x + tan x
4. sec²(sin x) cos x + tan² x

Answer 1

Correct Answer - Option 2 : sec²(sin x) cos x + sec x tan x

Concept:

Chain rule: \(\rm\frac{d}{d x}[f(g(x))]=f^{\prime}(g(x)) g^{\prime}(x)\)

\(\rm\frac{d}{d x}[\tan x]=sec^2 x\) , \(\rm\frac{d}{d x}[\sec x]=sec \ x \tan\ x\)

Calculation:

Here let, f(x) =  tan (sin x) + sec x

f'(x) = sec2(sin x) × \(\rm\frac{d}{d x}[\sin x]\) + sec x tan x

= sec2(sin x) cos x + sec x tan x

∴ The derivative of tan (sin x) is sec2(sin x) cos x + sec x tan x