Question

If f(x) = e^sin x, then what is derivative of f(x) ?
1. (sin x)e^{sin x}
2. (cos x)e^{sin x}
3. (sec x)e^{sin x}
4. (cosec x)e^{sin x}

Answer 1

Correct Answer - Option 2 : (cos x)e^{sin x}

Concept:

Derivative Rule:

Let y = ef(x) then

\(\rm \dfrac {dy}{dx} = e^{f(x)} \dfrac {d}{dx} f(x)\)

Calculations:

Given, f(x) = e^sin x

Taking derivative w. r. to x on both side, we get

⇒f'(x) = \(\rm \dfrac {d}{dx}\)(esin x)

⇒f'(x) = esin x \(\rm \dfrac {d}{dx} (\sin x)\)

⇒f'(x) = esin x \(\rm ( \cos x)\)

⇒f'(x) = \(\rm (\cos x)\)esin x