Given the function f(x)=(3sqrtx)/(2)-(3)/(sqrtx), find f^(')(6). Express your answer as a single fraction in simplest radical form. Answer: f^(')(6)=

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Given the function  f(x)=(3sqrtx)/(2)-(3)/(sqrtx), find  f^(')(6). Express your answer as a single fraction in simplest radical form. Answer:  f^(')(6)=

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Find Derivative of Function: First, we need to find the derivative of the function f(x) = (3√x)/(2) - (3)/(√x). To do this, we will use the power rule for differentiation. The square root of x can be written as x^(1/2), so the function becomes f(x) = (3/2)x^(1/2) - 3x^(-1/2).Differentiate First Term: Now, let's differentiate the first term (3/2)x^(1/2). Using the power rule, the derivative of x^(n) is n*x^(n-1), so the derivative of (3/2)x^(1/2) is (3/2)*(1/2)*x^(1/2 - 1) = (3/4)x^(-1/2).Differentiate Second Term: Next, we differentiate the second term -3x^(-1/2). Again, using the power rule, the derivative of -3x^(-1/2) is -3*(-1/2)*x^(-1/2 - 1) = (3/2)x^(-3/2).Combine Derivatives: Combining the derivatives of both terms, we get the derivative of the function f(x), which is f'(x) = (3/4)x^(-1/2) + (3/2)x^(-3/2).Evaluate at x=6: Now we need to evaluate the derivative at x=6. Substituting x with 6, we get f'(6) = (3/4)(6)^(-1/2) + (3/2)(6)^(-3/2).Simplify Square Root and Cube Root: To simplify, we calculate the square root and the cube root of 6. The square root of 6 is √6, and the cube root of 6 is 6^(3/2) = (√6)^3. So, f'(6) = (3/4)(1/√6) + (3/2)(1/((√6)^3)).Find Common Denominator: We can simplify the expression further by finding a common denominator. The common denominator for √6 and (√6)^3 is (√6)^3. So, we multiply the first term by √6/√6 to get the common denominator. f'(6) = (3/4)(√6/√6)(1/√6) + (3/2)(1/((√6)^3)) = (3√6)/(4√6^2) + (3/2)(1/√6^3).Combine Terms with Common Denominator: Simplifying the denominators, we have √6^2 = 6 and √6^3 = 6√6. So, f'(6) = (3√6)/(4*6) + (3/2)(1/(6√6)) = (3√6)/(24) + (3/2)(1/(6√6)).Simplify Fraction: Now, we combine the terms over a common denominator, which is 24√6. To do this, we multiply the second term by 4/4 to get the common denominator. f'(6) = (3√6)/(24) + (3/2)(4/(24√6)) = (3√6)/(24) + (6)/(24√6).Factor Out 3 from Numerator: Adding the two fractions together, we get f'(6) = (3√6 + 6)/(24√6). This is the derivative of the function f(x) evaluated at x=6.Finally, we simplify the fraction by factoring out a 3 from the numerator. f'(6) = 3(√6 + 2)/(24√6). We can simplify this further by dividing both the numerator and the denominator by 3. f'(6) = (√6 + 2)/(8√6).

Given the function $f(x)=\frac{3 \sqrt{x}}{2}-\frac{3}{\sqrt{x}}$ , find $f^{\prime}(6)$ . Express your answer as a single fraction in simplest radical form. $\newline$ Answer: $f^{\prime}(6)=$

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Given the function $f(x)=\frac{3 \sqrt{x}}{2}-\frac{3}{\sqrt{x}}$ , find $f^{\prime}(6)$ . Express your answer as a single fraction in simplest radical form. $\newline$ Answer: $f^{\prime}(6)=$