let y = e^{sin x} + (tan x)^x

⇒ y = a + b

where a= e^{sin x} ; b = (tan x)^x

a= e^{sin x}

Taking log both the sides:

⇒ log a= log e^{sin x}

⇒ log a= sin x log e

{log x^a = alog x}

⇒ log a= sin x {log e =1}

Differentiating with respect to x:

Put the value of a = e^{sin x}

b = (tan x)^x

Taking log both the sides:

⇒ log b= log (tan x)^x

⇒ log b= x log (tan x)

{log x^a = alog x}

Differentiating with respect to x:

Put the value of b = (tan x)^x :