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{:[g(x)=sqrt(sin(x))],[g^(')(x)=?]:} Choose 1 answer: (A) (cos(sqrtx))/(2sqrtx) (B) sqrt(cos(x)) (C) ([sin(x)]^(-(1)/(2)))/(2) (D) (cos(x))/(2sqrt(sin(x)))

g(x)=sin⁡(x)g′(x)=? \begin{array}{l} g(x)=\sqrt{\sin (x)} \\ g^{\prime}(x)=? \end{array} \newlineChoose 11 answer:\newline(A) cos⁡(x)2x \frac{\cos (\sqrt{x})}{2 \sqrt{x}} \newline(B) cos⁡(x) \sqrt{\cos (x)} \newline(C) [sin⁡(x)]−122 \frac{[\sin (x)]^{-\frac{1}{2}}}{2} \newline(D) cos⁡(x)2sin⁡(x) \frac{\cos (x)}{2 \sqrt{\sin (x)}}

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Q. g(x)=sin⁡(x)g′(x)=? \begin{array}{l} g(x)=\sqrt{\sin (x)} \\ g^{\prime}(x)=? \end{array} \newlineChoose 11 answer:\newline(A) cos⁡(x)2x \frac{\cos (\sqrt{x})}{2 \sqrt{x}} \newline(B) cos⁡(x) \sqrt{\cos (x)} \newline(C) [sin⁡(x)]−122 \frac{[\sin (x)]^{-\frac{1}{2}}}{2} \newline(D) cos⁡(x)2sin⁡(x) \frac{\cos (x)}{2 \sqrt{\sin (x)}}
  1. Identify function: Identify the function to differentiate.\newlineWe have the function g(x)=sin⁡(x)g(x) = \sqrt{\sin(x)}. We need to find its derivative, denoted as g′(x)g'(x).
  2. Apply chain rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is the square root function, and the inner function is the sine function.
  3. Differentiate outer function: Differentiate the outer function.\newlineThe outer function is u\sqrt{u}, where u=sin⁡(x)u = \sin(x). The derivative of u\sqrt{u} with respect to uu is 12u−12\frac{1}{2}u^{-\frac{1}{2}}.
  4. Differentiate inner function: Differentiate the inner function.\newlineThe inner function is sin⁡(x)\sin(x). The derivative of sin⁡(x)\sin(x) with respect to xx is cos⁡(x)\cos(x).
  5. Apply chain rule with derivatives: Apply the chain rule using the derivatives from steps 33 and 44.\newlineg′(x)=ddx[sin⁡(x)]=ddu[u]⋅ddx[sin⁡(x)]=12u−12⋅cos⁡(x)=12(sin⁡(x))−12⋅cos⁡(x)g'(x) = \frac{d}{dx}[\sqrt{\sin(x)}] = \frac{d}{du}[\sqrt{u}] \cdot \frac{d}{dx}[\sin(x)] = \frac{1}{2}u^{-\frac{1}{2}} \cdot \cos(x) = \frac{1}{2}(\sin(x))^{-\frac{1}{2}} \cdot \cos(x).
  6. Substitute uu back: Substitute uu back with sin⁡(x)\sin(x).g′(x)=12(sin⁡(x))−12⋅cos⁡(x)=cos⁡(x)2sin⁡(x)g'(x) = \frac{1}{2}(\sin(x))^{-\frac{1}{2}} \cdot \cos(x) = \frac{\cos(x)}{2\sqrt{\sin(x)}}.
  7. Match with given choices: Match the result with the given choices.\newlineThe correct answer is (D) cos⁡(x)2sin⁡(x)\frac{\cos(x)}{2\sqrt{\sin(x)}}.

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