Given the function f(x)=(8x^(3)-9x^(2)+9)(-9+2x^(2)), find f^(')(x) in any form. Answer: f^(')(x)=

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Given the function  f(x)=(8x^(3)-9x^(2)+9)(-9+2x^(2)), 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)=(8x^(3)-9x^(2)+9)(-9+2x^(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) = 8x^(3)-9x^(2)+9 and the second function as v(x) = -9+2x^(2). We need to find the derivatives u'(x) and v'(x) separately.Derivative of u(x): First, we find the derivative of u(x) = 8x^(3)-9x^(2)+9. Using the power rule, we get: u'(x) = d/dx(8x^(3)) - d/dx(9x^(2)) + d/dx(9) u'(x) = 3*8x^(3-1) - 2*9x^(2-1) + 0 u'(x) = 24x^(2) - 18xDerivative of v(x): Next, we find the derivative of v(x) = -9+2x^(2). Again, using the power rule, we get: v'(x) = d/dx(-9) + d/dx(2x^(2)) v'(x) = 0 + 2*2x^(2-1) v'(x) = 4xApply Product Rule: Now we apply the product rule: f'(x) = u'(x)v(x) + u(x)v'(x) f'(x) = (24x^(2) - 18x)(-9+2x^(2)) + (8x^(3)-9x^(2)+9)(4x)Expand Products: We need to expand the products: f'(x) = (24x^(2)*(-9) + 24x^(2)*(2x^(2)) - 18x*(-9) + 18x*(-2x^(2))) + (8x^(3)*4x - 9x^(2)*4x + 9*4x) f'(x) = (-216x^(2) + 48x^(4) + 162x - 36x^(3)) + (32x^(4) - 36x^(3) + 36x)Combine Like Terms: Now we combine like terms: f'(x) = 48x^(4) - 216x^(2) + 162x - 36x^(3) + 32x^(4) - 36x^(3) + 36x f'(x) = (48x^(4) + 32x^(4)) + (-36x^(3) - 36x^(3)) + (-216x^(2)) + (162x + 36x) f'(x) = 80x^(4) - 72x^(3) - 216x^(2) + 198x

Given the function $f(x)=\left(8 x^{3}-9 x^{2}+9\right)\left(-9+2 x^{2}\right)$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$

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Given the function f(x)=(8x3−9x2+9)(−9+2x2) f(x)=\left(8 x^{3}-9 x^{2}+9\right)\left(-9+2 x^{2}\right) f(x)=(8x3−9x2+9)(−9+2x2), find f′(x) f^{\prime}(x) f′(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)= f′(x)=

Given the function $f(x)=\left(8 x^{3}-9 x^{2}+9\right)\left(-9+2 x^{2}\right)$ , find $f^{\prime}(x)$ in any form. $\newline$ Answer: $f^{\prime}(x)=$