Given the function f(x)=(7-8x-10x^(3))(-6x^(2)-5), find f^(')(x) in any form. Answer: f^(')(x)=

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Given the function  f(x)=(7-8x-10x^(3))(-6x^(2)-5), find  f^(')(x) in any form. Answer:  f^(')(x)=

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Product Rule Explanation: To find the derivative of the function f(x)=(7-8x-10x^(3))(-6x^(2)-5), 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) = 7 - 8x - 10x^3 and the second function as v(x) = -6x^2 - 5. We will find the derivatives of u(x) and v(x) separately.Derivative of u(x): The derivative of u(x) = 7 - 8x - 10x^3 with respect to x is u'(x) = 0 - 8 - 30x^2. Simplifying, we get u'(x) = -8 - 30x^2.Derivative of v(x): The derivative of v(x) = -6x^2 - 5 with respect to x is v'(x) = -12x - 0. Simplifying, we get v'(x) = -12x.Apply Product Rule: Now, we apply the product rule: f'(x) = u'(x)v(x) + u(x)v'(x). Substituting the derivatives we found, we get f'(x) = (-8 - 30x^2)(-6x^2 - 5) + (7 - 8x - 10x^3)(-12x).Distribute Terms: We will now distribute the terms in the expression f'(x) = (-8 - 30x^2)(-6x^2 - 5) + (7 - 8x - 10x^3)(-12x). This involves multiplying each term in the first product and adding it to the result of multiplying each term in the second product.First Product Simplification: Multiplying the terms in the first product, we get: (-8)(-6x^2) + (-8)(-5) + (-30x^2)(-6x^2) + (-30x^2)(-5). This simplifies to 48x^2 + 40 - 180x^4 + 150x^2.Second Product Simplification: Multiplying the terms in the second product, we get: (7)(-12x) + (-8x)(-12x) + (-10x^3)(-12x). This simplifies to -84x + 96x^2 + 120x^4.Combine Results: Adding the results from the two products, we get f'(x) = (48x^2 + 40 - 180x^4 + 150x^2) + (-84x + 96x^2 + 120x^4).Correct v'(x): Combining like terms, we get f'(x) = -180x^4 + 120x^4 + 48x^2 + 150x^2 + 96x^2 + 40 - 84x. This simplifies to f'(x) = -60x^4 + 294x^2 - 84x + 40.Reapply Product Rule: Correcting the derivative of v(x), we have v'(x) = -12x. Now we will reapply the product rule with the correct derivatives: f'(x) = u'(x)v(x) + u(x)v'(x) = (-8 - 30x^2)(-6x^2 - 5) + (7 - 8x - 10x^3)(-12x).First Product with Correct Derivative: Multiplying the terms in the first product with the correct derivative, we get: (-8)(-6x^2) + (-8)(-5) + (-30x^2)(-6x^2) + (-30x^2)(-5). This simplifies to 48x^2 + 40 + 180x^4 - 150x^2.Second Product with Correct Derivative: Multiplying the terms in the second product with the correct derivative, we get: (7)(-12x) + (-8x)(-12x) + (-10x^3)(-12x). This simplifies to -84x + 96x^2 + 120x^4.Combine Results with Correct Derivatives: Adding the results from the two products with the correct derivatives, we get f'(x) = (48x^2 + 40 + 180x^4 - 150x^2) + (-84x + 96x^2 + 120x^4).Combining like terms with the correct derivatives, we get f'(x) = 180x^4 + 120x^4 + 48x^2 - 150x^2 + 96x^2 + 40 - 84x. This simplifies to f'(x) = 300x^4 - 6x^2 - 84x + 40.

Given the function $f(x)=\left(7-8 x-10 x^{3}\right)\left(-6 x^{2}-5\right)$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$

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Given the function f(x)=(7−8x−10x3)(−6x2−5) f(x)=\left(7-8 x-10 x^{3}\right)\left(-6 x^{2}-5\right) f(x)=(7−8x−10x3)(−6x2−5), find f′(x) f^{\prime}(x) f′(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)= f′(x)=

Given the function $f(x)=\left(7-8 x-10 x^{3}\right)\left(-6 x^{2}-5\right)$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$