Given the function f(x)=(-5-4x^(-2))(x^(2)+6+6x), find f^(')(x) in any form. Answer: f^(')(x)=

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Given the function  f(x)=(-5-4x^(-2))(x^(2)+6+6x), find  f^(')(x) in any form. Answer:  f^(')(x)=

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Product Rule Application: To find the derivative of the function f(x) = (-5 - 4x^(-2))(x^2 + 6 + 6x), 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.Derivative of u(x): Let's denote the two functions as u(x) = -5 - 4x^(-2) and v(x) = x^2 + 6 + 6x. We will first find the derivative of u(x) with respect to x. The derivative of a constant is 0, and the derivative of -4x^(-2) is 8x^(-3) using the power rule, which states that (d/dx)(x^n) = nx^(n-1). So, (d/dx)(u(x)) = (d/dx)(-5) + (d/dx)(-4x^(-2)) = 0 + 8x^(-3).Derivative of v(x): Next, we will find the derivative of v(x) with respect to x. The derivative of x^2 is 2x, the derivative of 6 is 0, and the derivative of 6x is 6 using the power rule and the fact that the derivative of a constant is 0. So, (d/dx)(v(x)) = (d/dx)(x^2) + (d/dx)(6) + (d/dx)(6x) = 2x + 0 + 6.Applying Product Rule to Find f'(x): Now we can apply the product rule to find f'(x). f'(x) = u'(x)v(x) + u(x)v'(x) f'(x) = (8x^(-3))(x^2 + 6 + 6x) + (-5 - 4x^(-2))(2x + 6).Simplifying f'(x) Expression: We will now simplify the expression for f'(x). First, distribute 8x^(-3) across (x^2 + 6 + 6x): 8x^(-3)x^2 + 8x^(-3)6 + 8x^(-3)6x = 8x^(-1) + 48x^(-3) + 48x^(-2).Next, distribute (-5 - 4x^(-2)) across (2x + 6): -5(2x) - 5(6) - 4x^(-2)(2x) - 4x^(-2)(6) = -10x - 30 - 8x^(-1) - 24x^(-2).Now we combine like terms to get the final expression for f'(x): f'(x) = 8x^(-1) + 48x^(-3) + 48x^(-2) - 10x - 30 - 8x^(-1) - 24x^(-2). f'(x) = -10x - 30 + 48x^(-3) + 24x^(-2) - 24x^(-2) + 48x^(-2).

Given the function $f(x)=\left(-5-4 x^{-2}\right)\left(x^{2}+6+6 x\right)$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$

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Given the function f(x)=(−5−4x−2)(x2+6+6x) f(x)=\left(-5-4 x^{-2}\right)\left(x^{2}+6+6 x\right) f(x)=(−5−4x−2)(x2+6+6x), find f′(x) f^{\prime}(x) f′(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)= f′(x)=

Given the function $f(x)=\left(-5-4 x^{-2}\right)\left(x^{2}+6+6 x\right)$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$