find the derivative of the function.\nf(x)=\\sqrt{5 + x\\sin x}\n\\frac{d}{dx}\\sqrt{5 + x\\sin x}=\\square

Explanation:

Step1: Rewrite the function

Rewrite ( f(x)=\sqrt{5 + x\sin x}=(5 + x\sin x)^{\frac{1}{2}} ).

Step2: Apply the chain rule

The chain rule states that if ( y = u^{\frac{1}{2}}) and (u = 5+x\sin x), then (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). First, find (\frac{dy}{du}): (\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}=\frac{1}{2\sqrt{5 + x\sin x}}). Then, find (\frac{du}{dx}) using the sum rule (((a + b)^\prime=a^\prime + b^\prime)) and the product rule (((uv)^\prime = u^\prime v+uv^\prime) where (u = x) and (v=\sin x)). (\frac{du}{dx}=\frac{d(5)}{dx}+\frac{d(x\sin x)}{dx}). Since (\frac{d(5)}{dx} = 0), and (\frac{d(x\sin x)}{dx}=\sin x+x\cos x) (by product rule: ((x)^\prime\sin x+x(\sin x)^\prime=\sin x + x\cos x)).

Step3: Multiply the two derivatives

(\frac{dy}{dx}=\frac{1}{2\sqrt{5 + x\sin x}}\cdot(\sin x + x\cos x)).

Answer:

(\frac{\sin x + x\cos x}{2\sqrt{5 + x\sin x}})