Given the function y=(2+3x)/(3-2x^(2)), find (dy)/(dx) in simplified form. Answer: (dy)/(dx)=

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Given the function  y=(2+3x)/(3-2x^(2)), find  (dy)/(dx) in simplified form. Answer:  (dy)/(dx)=

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Apply Quotient Rule: To find the derivative of the function y=(2+3x)/(3-2x^2) with respect to x, we will use the quotient rule. The quotient rule states that if you have a function that is the quotient of two functions, u(x)/v(x), then its derivative is given by (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2. Here, u(x) = 2 + 3x and v(x) = 3 - 2x^2.Find u(x) Derivative: First, we need to find the derivative of u(x) = 2 + 3x with respect to x. The derivative of a constant is 0, and the derivative of 3x with respect to x is 3. Therefore, u'(x) = 0 + 3 = 3.Find v(x) Derivative: Next, we need to find the derivative of v(x) = 3 - 2x^2 with respect to x. The derivative of a constant is 0, and the derivative of -2x^2 with respect to x is -4x. Therefore, v'(x) = 0 - 4x = -4x.Apply Quotient Rule: Now we apply the quotient rule. We have u(x) = 2 + 3x, v(x) = 3 - 2x^2, u'(x) = 3, and v'(x) = -4x. Plugging these into the quotient rule formula, we get:  (dy)/(dx) = ((3 - 2x^2) * 3 - (2 + 3x) * (-4x)) / (3 - 2x^2)^2.Simplify Numerator: Simplify the expression in the numerator:  (3 - 2x^2) * 3 - (2 + 3x) * (-4x) = 9 - 6x^2 + 8x + 12x^2.Combine Like Terms: Combine like terms in the numerator:  9 - 6x^2 + 8x + 12x^2 = 9 + 8x + 6x^2.Final Derivative: Now we have the simplified form of the derivative:  (dy)/(dx) = (9 + 8x + 6x^2) / (3 - 2x^2)^2.

Given the function $y=\frac{2+3 x}{3-2 x^{2}}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$

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Given the function $y=\frac{2+3 x}{3-2 x^{2}}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$