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Given the function f(x)=(3)/(2sqrtx)-(3sqrtx)/(2), find f^(')(3). Express your answer as a single fraction in simplest radical form. Answer: f^(')(3)=

Given the function f(x)=32x−3x2 f(x)=\frac{3}{2 \sqrt{x}}-\frac{3 \sqrt{x}}{2} , find f′(3) f^{\prime}(3) . Express your answer as a single fraction in simplest radical form.\newlineAnswer: f′(3)= f^{\prime}(3)=

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Q. Given the function f(x)=32x−3x2 f(x)=\frac{3}{2 \sqrt{x}}-\frac{3 \sqrt{x}}{2} , find f′(3) f^{\prime}(3) . Express your answer as a single fraction in simplest radical form.\newlineAnswer: f′(3)= f^{\prime}(3)=
  1. Rewrite function: First, we need to find the derivative of the function f(x)=32x−3x2f(x) = \frac{3}{2\sqrt{x}} - \frac{3\sqrt{x}}{2}. Let's start by rewriting the function in a form that makes it easier to differentiate.f(x)=32x12−32x12f(x) = \frac{3}{2x^{\frac{1}{2}}} - \frac{3}{2}x^{\frac{1}{2}}
  2. Differentiate function: Now, let's differentiate the function using the power rule, which states that the derivative of xnx^n with respect to xx is n∗x(n−1)n*x^{(n-1)}.\newlinef′(x)=ddx[32x(1/2)]−ddx[32x(1/2)]f'(x) = \frac{d}{dx}\left[\frac{3}{2x^{(1/2)}}\right] - \frac{d}{dx}\left[\frac{3}{2}x^{(1/2)}\right]\newlinef′(x)=(32)∗(−12)∗x(−3/2)−(32)∗(12)∗x(−1/2)f'(x) = \left(\frac{3}{2}\right) * \left(-\frac{1}{2}\right) * x^{(-3/2)} - \left(\frac{3}{2}\right) * \left(\frac{1}{2}\right) * x^{(-1/2)}
  3. Simplify derivative: Simplify the expression for the derivative. f′(x)=−(34)x(−32)−(34)x(−12)f'(x) = -(\frac{3}{4})x^{(-\frac{3}{2})} - (\frac{3}{4})x^{(-\frac{1}{2})}
  4. Evaluate at x=3x=3: Now, we need to evaluate the derivative at x=3x = 3.f′(3)=−(34)3(−32)−(34)3(−12)f'(3) = -(\frac{3}{4})3^{(-\frac{3}{2})} - (\frac{3}{4})3^{(-\frac{1}{2})}
  5. Calculate values: Calculate the values of 3−323^{-\frac{3}{2}} and 3−123^{-\frac{1}{2}}.\newline3−32=1332=133=127=1333^{-\frac{3}{2}} = \frac{1}{3^{\frac{3}{2}}} = \frac{1}{\sqrt{3^3}} = \frac{1}{\sqrt{27}} = \frac{1}{3\sqrt{3}}\newline3−12=1312=133^{-\frac{1}{2}} = \frac{1}{3^{\frac{1}{2}}} = \frac{1}{\sqrt{3}}
  6. Substitute values: Substitute these values into the expression for f′(3)f'(3).f′(3)=−(34)(133)−(34)(13)f'(3) = -\left(\frac{3}{4}\right)\left(\frac{1}{3\sqrt{3}}\right) - \left(\frac{3}{4}\right)\left(\frac{1}{\sqrt{3}}\right)
  7. Combine terms: Simplify the expression by combining the terms.\newlinef′(3)=−(14)(13)−(34)(13)f'(3) = -(\frac{1}{4})(\frac{1}{\sqrt{3}}) - (\frac{3}{4})(\frac{1}{\sqrt{3}})\newlinef′(3)=−143−343f'(3) = -\frac{1}{4\sqrt{3}} - \frac{3}{4\sqrt{3}}\newlinef′(3)=(−1−3)43f'(3) = \frac{(-1 - 3)}{4\sqrt{3}}\newlinef′(3)=−443f'(3) = -\frac{4}{4\sqrt{3}}
  8. Simplify expression: Simplify the fraction by canceling out common factors.\newlinef′(3)=−13f'(3) = -\frac{1}{\sqrt{3}}

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