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

$\begin{array}{l} g(x)=\sqrt{\sin (x)} \\ g^{\prime}(x)=? \end{array}$ $\newline$ Choose $1$ answer: $\newline$ (A) $\sqrt{\cos (x)}$ $\newline$ (B) $\frac{\cos (\sqrt{x})}{2 \sqrt{x}}$ $\newline$ (C) $\frac{\cos (x)}{2 \sqrt{\sin (x)}}$ $\newline$ (D) $\frac{[\sin (x)]^{-\frac{1}{2}}}{2}$

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Calculus

Find derivatives of using multiple formulae

Full solution

\begin{array}{l} g(x)=\sqrt{\sin (x)} \\ g^{\prime}(x)=? \end{array}

\newline

Choose

1

answer:

\newline

(A)

\sqrt{\cos (x)}

\newline

(B)

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

\newline

(C)

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

\newline

(D)

\frac{[\sin (x)]^{-\frac{1}{2}}}{2}

Apply Chain Rule: Apply the chain rule to differentiate $g(x) = \sqrt{\sin(x)}$ . $\newline$ 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. $\newline$ Let $u = \sin(x)$ , then $g(x) = \sqrt{u}$ .
Differentiate Outer Function: Differentiate the outer function with respect to $u$ , where the outer function is $\sqrt{u}$ . The derivative of $\sqrt{u}$ with respect to $u$ is $(1/2)u^{-1/2}$ . $(d/du)(\sqrt{u}) = (1/2)u^{-1/2}$
Differentiate Inner Function: Differentiate the inner function $u = \sin(x)$ with respect to $x$ . The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$ . $\frac{d}{dx}(\sin(x)) = \cos(x)$
Apply Chain Rule Multiplication: Apply the chain rule by multiplying the derivatives from Step $2$ and Step $3$ . $\newline$ $g'(x) = \frac{d}{du}(\sqrt{u}) \cdot \frac{d}{dx}(u)$ $\newline$ $g'(x) = \frac{1}{2}u^{-\frac{1}{2}} \cdot \cos(x)$ $\newline$ Since $u = \sin(x)$ , we substitute back to get: $\newline$ $g'(x) = \frac{1}{2}(\sin(x))^{-\frac{1}{2}} \cdot \cos(x)$
Simplify Expression: Simplify the expression for $g'(x)$ . $g'(x) = \frac{\cos(x)}{2\sqrt{\sin(x)}}$ This matches answer choice (C).