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

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

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Apply Differentiation Rules: To find the derivative of the function f(x) = sqrt(x) - 2sqrt(x^3), we need to apply the rules of differentiation to each term separately. The derivative of sqrt(x) with respect to x is (1/2)x^(-1/2). The derivative of sqrt(x^3) with respect to x is (3/2)x^(3/2 - 1) = (3/2)x^(1/2). Therefore, the derivative of f(x) is f'(x) = (1/2)x^(-1/2) - 2 * (3/2)x^(1/2).Derivative of sqrt(x): Now we simplify the expression for f'(x). f'(x) = (1/2)x^(-1/2) - 3x^(1/2).Derivative of sqrt(x^3): Next, we evaluate the derivative at x = 3. f'(3) = (1/2)3^(-1/2) - 3(3^(1/2)).Simplify f'(x): We calculate the values of 3^(-1/2) and 3^(1/2). 3^(-1/2) is the reciprocal of the square root of 3, which is 1/sqrt(3). 3^(1/2) is the square root of 3, which is sqrt(3). So, f'(3) = (1/2)(1/sqrt(3)) - 3(sqrt(3)).Evaluate at x = 3: We simplify the expression by finding a common denominator and combining the terms. f'(3) = (1/2)(1/sqrt(3)) - 3(sqrt(3)) can be written as (1 - 6sqrt(3))/(2sqrt(3)).Calculate 3^(-1/2) and 3^(1/2): To rationalize the denominator, we multiply the numerator and the denominator by sqrt(3). f'(3) = ((1 - 6sqrt(3))/(2sqrt(3))) * (sqrt(3)/sqrt(3)).Combine Terms: We perform the multiplication to rationalize the denominator. f'(3) = (sqrt(3) - 6*3)/(2*3).Rationalize Denominator: We simplify the expression further. f'(3) = (sqrt(3) - 18)/6.Perform Multiplication: Finally, we check if the expression is in its simplest form. Since there are no common factors to cancel out, the expression (sqrt(3) - 18)/6 is in its simplest form.

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

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Given the function f(x)=x−2x3 f(x)=\sqrt{x}-2 \sqrt{x^{3}} f(x)=x​−2x3​, find f′(3) f^{\prime}(3) f′(3). Express your answer as a single fraction in simplest radical form.\newlineAnswer: f′(3)= f^{\prime}(3)= f′(3)=

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