Given the function y=(-10x^(3)+5)(-4+10x^(2)+x^(3)), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

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Given the function  y=(-10x^(3)+5)(-4+10x^(2)+x^(3)), find  (dy)/(dx) in any form. Answer:  (dy)/(dx)=

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Define Functions: To find the derivative of the function y=(-10x^(3)+5)(-4+10x^(2)+x^(3)), we will use the product rule. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Let's denote the first function as f(x) = -10x^(3)+5 and the second function as g(x) = -4+10x^(2)+x^(3).Find f'(x): First, we find the derivative of f(x) with respect to x. The derivative of -10x^(3) is -30x^(2), and the derivative of a constant, 5, is 0. So, f'(x) = -30x^(2).Find g'(x): Next, we find the derivative of g(x) with respect to x. The derivative of -4 is 0, the derivative of 10x^(2) is 20x, and the derivative of x^(3) is 3x^(2). So, g'(x) = 20x + 3x^(2).Apply Product Rule: Now, we apply the product rule. The derivative of y with respect to x, denoted as (dy)/(dx), is given by: (dy)/(dx) = f'(x)g(x) + f(x)g'(x). Substituting the derivatives we found, we get: (dy)/(dx) = (-30x^(2))(-4+10x^(2)+x^(3)) + (-10x^(3)+5)(20x+3x^(2)).Expand Terms: We now expand the terms in the expression for (dy)/(dx): (dy)/(dx) = (-30x^(2))(-4) + (-30x^(2))(10x^(2)) + (-30x^(2))(x^(3)) + (-10x^(3))(20x) + (-10x^(3))(3x^(2)) + (5)(20x) + (5)(3x^(2)).Simplify Expression: Simplify the expression by multiplying the terms: (dy)/(dx) = 120x^(2) - 300x^(4) - 30x^(5) - 200x^(4) - 30x^(5) + 100x + 15x^(2).Combine Like Terms: Combine like terms in the expression: (dy)/(dx) = 100x + (120x^(2) + 15x^(2)) - (300x^(4) + 200x^(4)) - (30x^(5) + 30x^(5)). (dy)/(dx) = 100x + 135x^(2) - 500x^(4) - 60x^(5).

Given the function $y=\left(-10 x^{3}+5\right)\left(-4+10 x^{2}+x^{3}\right)$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$

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Given the function y=(−10x3+5)(−4+10x2+x3) y=\left(-10 x^{3}+5\right)\left(-4+10 x^{2}+x^{3}\right) y=(−10x3+5)(−4+10x2+x3), find dydx \frac{d y}{d x} dxdy​ in any form.\newlineAnswer: dydx= \frac{d y}{d x}= dxdy​=

Given the function $y=\left(-10 x^{3}+5\right)\left(-4+10 x^{2}+x^{3}\right)$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$