Given the function f(x)=(-10-2x^(2)+5x^(-1))(-8-x^(2)), find f^(')(x) in any form. Answer: f^(')(x)=

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Given the function  f(x)=(-10-2x^(2)+5x^(-1))(-8-x^(2)), find  f^(')(x) in any form. Answer:  f^(')(x)=

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Apply Product Rule: To find the derivative of the function f(x) = (-10 - 2x^2 + 5x^(-1))(-8 - x^2), 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.Define Functions: Let's denote the first function as u(x) = -10 - 2x^2 + 5x^(-1) and the second function as v(x) = -8 - x^2. We will first find the derivative of u(x), which is u'(x).Find u'(x): The derivative of u(x) = -10 - 2x^2 + 5x^(-1) is found by differentiating each term separately. The derivative of a constant is 0, the derivative of -2x^2 is -4x, and the derivative of 5x^(-1) is -5x^(-2) (using the power rule). So, u'(x) = 0 - 4x - 5x^(-2).Find v'(x): Now we will find the derivative of v(x) = -8 - x^2. The derivative of a constant is 0, and the derivative of -x^2 is -2x. So, v'(x) = 0 - 2x = -2x.Use Product Rule: Using the product rule, the derivative of f(x) is f'(x) = u'(x)v(x) + u(x)v'(x). Substituting the derivatives we found, we get f'(x) = (-4x - 5x^(-2))(-8 - x^2) + (-10 - 2x^2 + 5x^(-1))(-2x).Expand Expressions: Now we will expand the expressions to simplify them. For the first part, we distribute (-4x - 5x^(-2)) across (-8 - x^2), and for the second part, we distribute (-10 - 2x^2 + 5x^(-1)) across (-2x).Expand First Part: Expanding the first part, we get: (-4x)(-8) + (-4x)(-x^2) + (-5x^(-2))(-8) + (-5x^(-2))(-x^2) = 32x + 4x^3 + 40x^(-2) + 5x^(-2 + 2) = 32x + 4x^3 + 40x^(-2) + 5Expand Second Part: Expanding the second part, we get: (-10)(-2x) + (-2x^2)(-2x) + (5x^(-1))(-2x) = 20x - 4x^3 - 10Combine Like Terms: Now we combine like terms from both expanded parts to get the final derivative: f'(x) = (32x + 4x^3 + 40x^(-2) + 5) + (20x - 4x^3 - 10) = 32x + 4x^3 + 40x^(-2) + 5 + 20x - 4x^3 - 10 = 52x + 40x^(-2) - 5Final Derivative: We have found the derivative of the function f(x), which is: f'(x) = 52x + 40x^(-2) - 5 This is the final answer.

Given the function $f(x)=\left(-10-2 x^{2}+5 x^{-1}\right)\left(-8-x^{2}\right)$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$

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Given the function f(x)=(−10−2x2+5x−1)(−8−x2) f(x)=\left(-10-2 x^{2}+5 x^{-1}\right)\left(-8-x^{2}\right) f(x)=(−10−2x2+5x−1)(−8−x2), find f′(x) f^{\prime}(x) f′(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)= f′(x)=

Given the function $f(x)=\left(-10-2 x^{2}+5 x^{-1}\right)\left(-8-x^{2}\right)$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$