Let g(x)=sqrtxsin(x). g^(')(x)=

Identify Components: Identify the function components to apply the product rule for differentiation. g(x) = sqrt(x)sin(x) = (x^(1/2))sin(x)Apply Product Rule: Differentiate using the product rule: (fg)^(') = f^(')g + fg^('). Let f(x) = x^(1/2) and g(x) = sin(x).Differentiate f(x): Differentiate f(x) = x^(1/2). f^(')(x) = (1/2)x^(-1/2)Differentiate g(x): Differentiate g(x) = sin(x). g^(')(x) = cos(x)Apply Rule: Apply the product rule. g^(')(x) = f^(')(x)g(x) + f(x)g^(')(x) g^(')(x) = (1/2)x^(-1/2)sin(x) + x^(1/2)cos(x)Simplify Expression: Simplify the expression. g^(')(x) = (1/2)x^(-1/2)sin(x) + x^(1/2)cos(x)

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Let $g(x)=\sqrt{x} \sin (x)$ . $\newline$ $g^{\prime}(x)=$

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Algebra 1

Domain and range of square root functions: equations

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Q. Let

g(x)=\sqrt{x} \sin (x)

\newline

g^{\prime}(x)=

Identify Components: Identify the function components to apply the product rule for differentiation. $g(x) = \sqrt{x}\sin(x) = (x^{1/2})\sin(x)$
Apply Product Rule: Differentiate using the product rule: $fg$ ^{'} = f^{'}g + fg^{'}\. Let $f(x) = x^{\frac{1}{2}}$ and $g(x) = \sin(x)$ .
Differentiate $f(x)$ : Differentiate $f(x) = x^{(1/2)}$ . $\newline$ $f^{\prime}(x) = \left(\frac{1}{2}\right)x^{(-1/2)}$
Differentiate $g(x)$ : Differentiate $g(x) = \sin(x)$ . $g^{\prime}(x) = \cos(x)$
Apply Rule: Apply the product rule. $\newline$ $g^{\prime}(x) = f^{\prime}(x)g(x) + f(x)g^{\prime}(x)$ $\newline$ $g^{\prime}(x) = \left(\frac{1}{2}\right)x^{-\frac{1}{2}}\sin(x) + x^{\frac{1}{2}}\cos(x)$
Simplify Expression: Simplify the expression. $g^{'}(x) = \frac{1}{2}x^{-\frac{1}{2}}\sin(x) + x^{\frac{1}{2}}\cos(x)$