What is the local linearization of e^sin(x) near x=1?

Answer 1

Take the taylor series(lineal part)
f(x) = f(1) +(x-1)f´(1)
f´(x) = e^sin x *cos x and f´(1) = e^sin1*cos1
so
e^sinx = e^sin 1+(x-1) (e^sin1 *cos 1)