Question

Find the derivative of log (sin e^x)
1. e^- x ⋅ tan e^x
2. tan e^2x
3. -e^x ⋅ tan e^x
4. e^x ⋅ tan e^x

Answer 1

Correct Answer - Option 4 : e^x ⋅ tan e^x

Concept:

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

\(\rm\frac{d}{d x}[\sin x]=cos x\)

\(\rm\frac{d}{d x}[\ e^x]=e^x\)

\(\rm\frac{d}{d x}[\log x]= {1\over x}\)

Calculation:

Given: f(x) =  log (sin e^x)

f'(x) = \(\rm 1\over {\sin e^x}\) ⋅ \(\rm\frac{d}{d x}[\sin e^x]\)

= \(\rm 1\over {\sin e^x}\)⋅ (cos e^x) ⋅ \(\rm\frac{d}{d x}(e^x)\)

= \(\rm {{\cos e^x}\over {\sin e^x}} ⋅ e^x\)              (∵ \(\rm {{\cos x}\over {\sin x}} = \tan x\))

= e^x⋅ tan e^x