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Given the function f(x)=(-1-10 x+5x^(-2))(6x^(-2)-9), find f^(')(x) in any form. Answer: f^(')(x)=

Given the function f(x)=(−1−10x+5x−2)(6x−2−9) f(x)=\left(-1-10 x+5 x^{-2}\right)\left(6 x^{-2}-9\right) , find f′(x) f^{\prime}(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)=

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Q. Given the function f(x)=(−1−10x+5x−2)(6x−2−9) f(x)=\left(-1-10 x+5 x^{-2}\right)\left(6 x^{-2}-9\right) , find f′(x) f^{\prime}(x) in any form.\newlineAnswer: f′(x)= f^{\prime}(x)=
  1. Product Rule Explanation: To find the derivative of the function f(x)f(x), we will use the product rule, which 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. The product rule is given by (u⋅v)′=u′⋅v+u⋅v′(u \cdot v)' = u' \cdot v + u \cdot v', where uu and vv are functions of xx.
  2. Identifying Functions: First, let's identify the two functions that are being multiplied:\newlineu(x)=−1−10x+5x−2u(x) = -1 - 10x + 5x^{-2}\newlinev(x)=6x−2−9v(x) = 6x^{-2} - 9\newlineNow we will find the derivatives of u(x)u(x) and v(x)v(x) separately.
  3. Derivative of u(x)u(x): The derivative of u(x)u(x) with respect to xx is:\newlineu′(x)=ddx(−1)−ddx(10x)+ddx(5x−2)u'(x) = \frac{d}{dx}(-1) - \frac{d}{dx}(10x) + \frac{d}{dx}(5x^{-2})\newlineu′(x)=0−10+ddx(5⋅x−2)u'(x) = 0 - 10 + \frac{d}{dx}(5 \cdot x^{-2})\newlineu′(x)=−10−10⋅x−3u'(x) = -10 - 10 \cdot x^{-3}
  4. Derivative of v(x)v(x): The derivative of v(x)v(x) with respect to xx is: v′(x)=ddx(6x−2)−ddx(9)v'(x) = \frac{d}{dx}(6x^{-2}) - \frac{d}{dx}(9) v′(x)=−12⋅x−3−0v'(x) = -12 \cdot x^{-3} - 0 v′(x)=−12⋅x−3v'(x) = -12 \cdot x^{-3}
  5. Applying Product Rule: Now, we apply the product rule to find f′(x)f'(x):f′(x)=u′(x)⋅v(x)+u(x)⋅v′(x)f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)f′(x)=(−10−10⋅x−3)⋅(6x−2−9)+(−1−10x+5x−2)⋅(−12⋅x−3)f'(x) = (-10 - 10 \cdot x^{-3}) \cdot (6x^{-2} - 9) + (-1 - 10x + 5x^{-2}) \cdot (-12 \cdot x^{-3})
  6. Simplifying Expression: We simplify the expression by multiplying the terms: f′(x)=(−10×6x−2+10×9)+(10×6x−5−10×9x−3)+(12x−3+120x−2−60x−5)f'(x) = (-10 \times 6x^{-2} + 10 \times 9) + (10 \times 6x^{-5} - 10 \times 9x^{-3}) + (12x^{-3} + 120x^{-2} - 60x^{-5})
  7. Combine Like Terms: Combine like terms and simplify the expression:\newlinef′(x)=(−60x−2+90)+(60x−5−90x−3)+(12x−3+120x−2−60x−5)f'(x) = (-60x^{-2} + 90) + (60x^{-5} - 90x^{-3}) + (12x^{-3} + 120x^{-2} - 60x^{-5})\newlinef′(x)=−60x−2+90+60x−5−90x−3+12x−3+120x−2−60x−5f'(x) = -60x^{-2} + 90 + 60x^{-5} - 90x^{-3} + 12x^{-3} + 120x^{-2} - 60x^{-5}\newlinef′(x)=60x−2+2x−3−60x−5f'(x) = 60x^{-2} + 2x^{-3} - 60x^{-5}
  8. Final Derivative: Finally, we write the derivative in a more conventional form by using positive exponents:\newlinef′(x)=60x2+2x3−60x5f'(x) = \frac{60}{x^2} + \frac{2}{x^3} - \frac{60}{x^5}

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