Correct Answer - Option 2 : sec
2(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