Question

let ( g(x)=sqrt{x} sin (x) ).
( g^{prime}(x)= )

Explanation:

Step1: Apply the product rule

The product rule states that if \(y = u\cdot v\), then \(y^\prime=u^\prime v + uv^\prime\). Let \(u = \sqrt{x}=x^{\frac{1}{2}}\) and \(v=\sin(x)\).
First, find \(u^\prime\): Using the power rule \((x^n)^\prime=nx^{n - 1}\), so \(u^\prime=\frac{1}{2}x^{\frac{1}{2}- 1}=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}\).
Second, find \(v^\prime\): The derivative of \(\sin(x)\) is \(\cos(x)\), so \(v^\prime=\cos(x)\).

Step2: Substitute into the product rule formula

\(g^\prime(x)=u^\prime v+uv^\prime=\frac{1}{2\sqrt{x}}\sin(x)+\sqrt{x}\cos(x)\)

Answer:

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