Let y=(5-3x)/(x^(2)+3x). (dy)/(dx)=

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Let  y=(5-3x)/(x^(2)+3x).  (dy)/(dx)=

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Given Function: Given the function y=(5-3x)/(x^2+3x), we need to find the derivative of y with respect to x, denoted as (dy)/(dx). To do this, we will use the quotient rule for differentiation, which states that if we have a function in the form of u(x)/v(x), then its derivative is given by (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2. Here, u(x) = 5 - 3x and v(x) = x^2 + 3x.Derivative of u(x): First, we need to find the derivative of u(x) with respect to x. The function u(x) = 5 - 3x is a linear function, and its derivative u'(x) is the coefficient of x, which is -3. u'(x) = -3Derivative of v(x): Next, we need to find the derivative of v(x) with respect to x. The function v(x) = x^2 + 3x is a polynomial, and its derivative v'(x) is found by differentiating each term separately. The derivative of x^2 is 2x, and the derivative of 3x is 3. v'(x) = 2x + 3Apply Quotient Rule: Now we apply the quotient rule. We have u(x) = 5 - 3x, u'(x) = -3, v(x) = x^2 + 3x, and v'(x) = 2x + 3. Plugging these into the quotient rule formula, we get: (dy)/(dx) = ((x^2 + 3x) * (-3) - (5 - 3x) * (2x + 3)) / (x^2 + 3x)^2Simplify Numerator: Let's simplify the numerator of the derivative expression: (dy)/(dx) = (-3x^2 - 9x) - (10x + 15 - 6x^2 - 9x) / (x^2 + 3x)^2 (dy)/(dx) = (-3x^2 - 9x - 10x - 15 + 6x^2 + 9x) / (x^2 + 3x)^2 (dy)/(dx) = (3x^2 - 10x - 15) / (x^2 + 3x)^2Final Derivative: Finally, we have the derivative of y with respect to x in its simplified form: (dy)/(dx) = (3x^2 - 10x - 15) / (x^2 + 3x)^2

Let $y=\frac{5-3 x}{x^{2}+3 x}$ . $\newline$ $\frac{d y}{d x}=$

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