Question

Differentiate y=3e^sinx

Answer 1

Solution:

Consider,

$\rm \longrightarrow y = 3 {e}^{ \sin(x)}$

Differentiating both sides w.r.t. x

$\rm \longrightarrow \dfrac{dy}{dx} = \dfrac{d}{dx} ( 3 {e}^{ \sin(x)} )$

$\rm \longrightarrow \dfrac{dy}{dx} =3 \times \dfrac{d}{dx} ( {e}^{ \sin(x)} )$

Now, we know that:

• $\boxed{\dfrac{d}{dx}(e^x) = e^x}$
• $\boxed{\dfrac{d}{dx}(\sin(x)) = \cos(x)}$

Now apply chain rule by using these formulas will give us:

$\rm \longrightarrow \dfrac{dy}{dx} =3 \times {e}^{ \sin(x)} \times \dfrac{d}{dx} ( \sin x)$

$\rm \longrightarrow \dfrac{dy}{dx} =3 \times {e}^{ \sin(x)} \times \cos x$

$\underline{ \underline{ \rm \implies \dfrac{dy}{dx} =3 \cos (x ) {e}^{ \sin(x)} }}$

This is the required answer.

Additional Information:

Chain rule

If we have some composite functions, then we use chain rule to find the derivative.

Let's say the function we are provided with is f(g(x)), then firstly we have to differentiate f(x) and then have to multiply it by the derivative of inner function which is g(x).