Given the function f(x)=(x^(3)-2)/(2x-3), find f^(')(x) in simplified form. Answer: f^(')(x)=

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Given the function  f(x)=(x^(3)-2)/(2x-3), find  f^(')(x) in simplified form. Answer:  f^(')(x)=

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Quotient Rule Explanation: To find the derivative of the function f(x) = (x^3 - 2) / (2x - 3), we will use the quotient rule. The quotient rule states that if we have a function that is the quotient of two functions, u(x) / v(x), then its derivative f'(x) is given by (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2. Here, u(x) = x^3 - 2 and v(x) = 2x - 3.Derivative of u(x): First, we need to find the derivative of u(x) = x^3 - 2. The derivative of x^3 is 3x^2, and the derivative of a constant is 0. Therefore, u'(x) = 3x^2.Derivative of v(x): Next, we need to find the derivative of v(x) = 2x - 3. The derivative of 2x is 2, and the derivative of a constant is 0. Therefore, v'(x) = 2.Applying Quotient Rule: Now we apply the quotient rule. We have u'(x) = 3x^2 and v'(x) = 2, so we plug these into the quotient rule formula:  f'(x) = ((2x - 3) * (3x^2) - (x^3 - 2) * (2)) / (2x - 3)^2.  Let's simplify this expression step by step.Multiplying (2x - 3) by 3x^2: First, we multiply (2x - 3) by 3x^2 to get 6x^3 - 9x^2.Multiplying (x^3 - 2) by 2: Next, we multiply (x^3 - 2) by 2 to get 2x^3 - 4.Subtracting Products: Now we subtract the second product from the first product to get the numerator of the derivative:  (6x^3 - 9x^2) - (2x^3 - 4) = 6x^3 - 9x^2 - 2x^3 + 4.Combining Like Terms: Simplify the numerator by combining like terms:  6x^3 - 2x^3 - 9x^2 + 4 = 4x^3 - 9x^2 + 4.Final Derivative: Now we have the simplified numerator, and we place it over the square of the denominator:  f'(x) = (4x^3 - 9x^2 + 4) / (2x - 3)^2.  This is the derivative of the function in simplified form.

Given the function $f(x)=\frac{x^{3}-2}{2 x-3}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$

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Given the function $f(x)=\frac{x^{3}-2}{2 x-3}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$