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Given the function y=(-10+7x^(-2)+x)(4x^(-2)+3), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

Given the function y=(−10+7x−2+x)(4x−2+3) y=\left(-10+7 x^{-2}+x\right)\left(4 x^{-2}+3\right) , find dydx \frac{d y}{d x} in any form.\newlineAnswer: dydx= \frac{d y}{d x}=

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Q. Given the function y=(−10+7x−2+x)(4x−2+3) y=\left(-10+7 x^{-2}+x\right)\left(4 x^{-2}+3\right) , find dydx \frac{d y}{d x} in any form.\newlineAnswer: dydx= \frac{d y}{d x}=
  1. Identify Function: Identify the function to differentiate.\newlineWe are given the function y=(−10+7x−2+x)(4x−2+3)y = (-10 + 7x^{-2} + x)(4x^{-2} + 3). We need to find the derivative of this function with respect to xx, which is denoted as dydx\frac{dy}{dx}.
  2. Apply Product Rule: Apply the product rule for differentiation.\newlineThe 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. Let's denote the first function as u(x)=−10+7x−2+xu(x) = -10 + 7x^{-2} + x and the second function as v(x)=4x−2+3v(x) = 4x^{-2} + 3.
  3. Differentiate u(x)u(x): Differentiate u(x)u(x) with respect to xx.
    u(x)=−10+7x−2+xu(x) = -10 + 7x^{-2} + x
    The derivative of u(x)u(x), denoted as u′(x)u'(x), is found by differentiating each term separately:
    u′(x)=0−14x−3+1u'(x) = 0 - 14x^{-3} + 1
    u′(x)=1−14x−3u'(x) = 1 - 14x^{-3}
  4. Differentiate v(x)v(x): Differentiate v(x)v(x) with respect to xx.
    v(x)=4x−2+3v(x) = 4x^{-2} + 3
    The derivative of v(x)v(x), denoted as v′(x)v'(x), is found by differentiating each term separately:
    v′(x)=−8x−3+0v'(x) = -8x^{-3} + 0
    v′(x)=−8x−3v'(x) = -8x^{-3}
  5. Apply Product Rule: Apply the product rule using the derivatives from steps 33 and 44.\newline(dy)/(dx)=u′(x)v(x)+u(x)v′(x)(dy)/(dx) = u'(x)v(x) + u(x)v'(x)\newlineSubstitute the derivatives and original functions:\newline(dy)/(dx)=(1−14x−3)(4x−2+3)+(−10+7x−2+x)(−8x−3)(dy)/(dx) = (1 - 14x^{-3})(4x^{-2} + 3) + (-10 + 7x^{-2} + x)(-8x^{-3})
  6. Expand Expression: Expand the expression to simplify.\newline(dy)/(dx)=(1)(4x−2)+(1)(3)−(14x−3)(4x−2)−(14x−3)(3)+(−10)(−8x−3)+(7x−2)(−8x−3)+(x)(−8x−3)(dy)/(dx) = (1)(4x^{-2}) + (1)(3) - (14x^{-3})(4x^{-2}) - (14x^{-3})(3) + (-10)(-8x^{-3}) + (7x^{-2})(-8x^{-3}) + (x)(-8x^{-3})\newline(dy)/(dx)=4x−2+3−56x−5−42x−3+80x−3−56x−5−8x−2(dy)/(dx) = 4x^{-2} + 3 - 56x^{-5} - 42x^{-3} + 80x^{-3} - 56x^{-5} - 8x^{-2}
  7. Combine Like Terms: Combine like terms.\newlinedydx=4x−2−8x−2+3−42x−3+80x−3−56x−5−56x−5\frac{dy}{dx} = 4x^{-2} - 8x^{-2} + 3 - 42x^{-3} + 80x^{-3} - 56x^{-5} - 56x^{-5}\newlinedydx=−4x−2+38x−3−112x−5+3\frac{dy}{dx} = -4x^{-2} + 38x^{-3} - 112x^{-5} + 3
  8. Rewrite in Standard Form: Rewrite the expression in a more standard form.\newlinedydx=−4x2+38x3−112x5+3\frac{dy}{dx} = -\frac{4}{x^2} + \frac{38}{x^3} - \frac{112}{x^5} + 3

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