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

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Given the function  y=(3x+2)/(2-x^(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=(3x+2)/(2-x^(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) = 3x + 2 and v(x) = 2 - x^2.Find u'(x): First, we need to find the derivative of u(x) = 3x + 2 with respect to x. The derivative of 3x is 3, and the derivative of a constant is 0, so u'(x) = 3.Find v'(x): Next, we need to find the derivative of v(x) = 2 - x^2 with respect to x. The derivative of 2 is 0, and the derivative of -x^2 is -2x, so v'(x) = -2x.Plug into Quotient Rule: Now we apply the quotient rule. We have u(x) = 3x + 2, v(x) = 2 - x^2, u'(x) = 3, and v'(x) = -2x. Plugging these into the quotient rule formula, we get:  (dy)/(dx) = ((2 - x^2) * 3 - (3x + 2) * (-2x)) / (2 - x^2)^2.Simplify Numerator: Simplify the expression in the numerator:  (dy)/(dx) = (6 - 3x^2 + 6x^2 + 4x) / (2 - x^2)^2.Combine Like Terms: Combine like terms in the numerator:  (dy)/(dx) = (6 + 4x + 3x^2) / (2 - x^2)^2.Final Derivative: The expression is now simplified, and we have found the derivative of y with respect to x:  (dy)/(dx) = (3x^2 + 4x + 6) / (2 - x^2)^2.

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

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