Let y=(4x^(2)+3x)/(2x-7). (dy)/(dx)=

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Let  y=(4x^(2)+3x)/(2x-7).  (dy)/(dx)=

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Quotient Rule Explanation: To find the derivative of the function y 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) = 4x^2 + 3x and v(x) = 2x - 7.Derivative of u(x): First, we need to find the derivative of u(x) with respect to x. The derivative of u(x) = 4x^2 + 3x is u'(x) = 8x + 3, using the power rule which states that the derivative of x^n is n*x^(n-1).Derivative of v(x): Next, we need to find the derivative of v(x) with respect to x. The derivative of v(x) = 2x - 7 is v'(x) = 2, since the derivative of a constant is 0 and the derivative of 2x is 2.Applying Quotient Rule: Now we apply the quotient rule. We have u'(x) = 8x + 3 and v'(x) = 2, so we plug these into the quotient rule formula:  (dy/dx) = ((2x - 7) * (8x + 3) - (4x^2 + 3x) * 2) / (2x - 7)^2.Simplifying Numerator: We simplify the expression in the numerator:  (2x - 7) * (8x + 3) = 16x^2 + 6x - 56x - 21, (4x^2 + 3x) * 2 = 8x^2 + 6x.  So the numerator becomes:  16x^2 + 6x - 56x - 21 - (8x^2 + 6x) = 16x^2 + 6x - 56x - 21 - 8x^2 - 6x.Further Simplification: Simplify the terms in the numerator:  16x^2 - 8x^2 + 6x - 6x - 56x - 21 = 8x^2 - 56x - 21.Now we have the simplified form of the derivative:  (dy/dx) = (8x^2 - 56x - 21) / (2x - 7)^2.

Let $y=\frac{4 x^{2}+3 x}{2 x-7}$ . $\newline$ $\frac{d y}{d x}=$

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Let $y=\frac{4 x^{2}+3 x}{2 x-7}$ . $\newline$ $\frac{d y}{d x}=$