`f(t) = sin(e^t) + e^sin(t)` Find the derivative of the function.

Textbook Question

Chapter 3, 3.4 - Problem 12 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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hkj1385 | (Level 1) Assistant Educator

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Note:- 1) If y = sin(at)  ; then dy/dx = a*cos(at)

2) If y = e^(at) ; then dy/dx = a*e^(at); where a = constant

Now, 

f(t) = sin(e^t) + e^(sint)

f(t) is in the form of u + v ; where u = sin(e^t) & v = e^(sint)

Thus, by the addtion rule of differentiation;

f'(t) = u' + v'

Now, u = sin(e^t)

thus, u' = (e^t)*cos(e^t)........(1)

Also, v = e^(sint)

Thus, v' = cost*e^(sint).......(2)

Hence, f'(x) = u' + v' = [(e^t)*cos(e^t)] + [cost*e^(sint)]

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