(2x^(3)-3x^(2)+4x+2)/((x-4)^(2)(x+3)^(2))

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Quotient Rule Explanation: To find the derivative of the given function, we will use the quotient rule, which states that the derivative of a function in the form of f(x)/g(x) is given by (f'(x)g(x) - f(x)g'(x))/(g(x))^2. The function we are differentiating is f(x) = 2x^3 - 3x^2 + 4x + 2 and g(x) = (x-4)^2(x+3)^2.Derivative of Numerator: First, we need to find the derivative of the numerator, f(x) = 2x^3 - 3x^2 + 4x + 2. Using the power rule, we get: f'(x) = d/dx(2x^3) - d/dx(3x^2) + d/dx(4x) + d/dx(2)        = 6x^2 - 6x + 4 + 0        = 6x^2 - 6x + 4Derivative of Denominator: Next, we need to find the derivative of the denominator, g(x) = (x-4)^2(x+3)^2. We will use the product rule and the chain rule. The product rule states that the derivative of a product h(x) = u(x)v(x) is given by h'(x) = u'(x)v(x) + u(x)v'(x). The chain rule states that the derivative of a composite function h(x) = f(g(x)) is h'(x) = f'(g(x))g'(x).Chain Rule Application: Let's find the derivative of (x-4)^2 and (x+3)^2 separately using the chain rule. For (x-4)^2, let u = x-4, then u^2 = (x-4)^2. The derivative is: d/dx(u^2) = 2u * d/dx(u)            = 2(x-4) * 1            = 2x - 8 For (x+3)^2, let v = x+3, then v^2 = (x+3)^2. The derivative is: d/dx(v^2) = 2v * d/dx(v)            = 2(x+3) * 1            = 2x + 6Product Rule Application: Now we apply the product rule to find the derivative of g(x) = (x-4)^2(x+3)^2. Let u = (x-4)^2 and v = (x+3)^2, then: g'(x) = u'(x)v(x) + u(x)v'(x)        = (2x - 8)(x+3)^2 + (x-4)^2(2x + 6)Simplify Derivative of Denominator: We need to expand the terms in g'(x) to simplify the expression: g'(x) = (2x - 8)(x^2 + 6x + 9) + (x^2 - 8x + 16)(2x + 6)        = (2x^3 + 12x^2 + 18x - 8x^2 - 48x - 72) + (2x^3 - 16x^2 + 32x + 6x^2 - 48x + 96)        = 2x^3 + 4x^2 - 30x - 72 + 2x^3 - 10x^2 - 16x + 96        = 4x^3 - 6x^2 - 46x + 24Apply Quotient Rule: Now we have both the derivative of the numerator f'(x) = 6x^2 - 6x + 4 and the derivative of the denominator g'(x) = 4x^3 - 6x^2 - 46x + 24. We can apply the quotient rule to find the derivative of the entire function: h'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2        = ((6x^2 - 6x + 4)((x-4)^2(x+3)^2) - (2x^3 - 3x^2 + 4x + 2)(4x^3 - 6x^2 - 46x + 24)) / ((x-4)^4(x+3)^4)Final Derivative Expression: The expression for h'(x) is quite complex, and expanding it fully would be very tedious and prone to errors. Instead, we can leave the derivative in its factored form, as simplifying it further is not required unless specifically asked for. Therefore, the derivative of the function (2x^3 - 3x^2 + 4x + 2) / ((x-4)^2(x+3)^2) with respect to x is: h'(x) = ((6x^2 - 6x + 4)((x-4)^2(x+3)^2) - (2x^3 - 3x^2 + 4x + 2)(4x^3 - 6x^2 - 46x + 24)) / ((x-4)^4(x+3)^4)

$\frac{2x^{3}-3x^{2}+4x+2}{(x-4)^{2}(x+3)^{2}}$

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$\frac{2x^{3}-3x^{2}+4x+2}{(x-4)^{2}(x+3)^{2}}$