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Given the function y=(-4+5x^(-3)+7x^(-2))(-2-5x^(-1)), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

Given the function y=(−4+5x−3+7x−2)(−2−5x−1) y=\left(-4+5 x^{-3}+7 x^{-2}\right)\left(-2-5 x^{-1}\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=(−4+5x−3+7x−2)(−2−5x−1) y=\left(-4+5 x^{-3}+7 x^{-2}\right)\left(-2-5 x^{-1}\right) , find dydx \frac{d y}{d x} in any form.\newlineAnswer: dydx= \frac{d y}{d x}=
  1. Identify Functions: We are given the function y=(−4+5x−3+7x−2)(−2−5x−1)y=(-4+5x^{-3}+7x^{-2})(-2-5x^{-1}) and we need to find its derivative with respect to xx. To do this, 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.
  2. Find Derivative of u: First, let's identify the two functions that are being multiplied. We have u=(−4+5x−3+7x−2)u=(-4+5x^{-3}+7x^{-2}) and v=(−2−5x−1)v=(-2-5x^{-1}). We will need to find the derivatives u′u' and v′v' separately.
  3. Find Derivative of v: Let's find the derivative of uu with respect to xx. We have u=(−4+5x−3+7x−2)u=(-4+5x^{-3}+7x^{-2}). The derivative of a constant is 00, and the derivative of xnx^n with respect to xx is n⋅x(n−1)n\cdot x^{(n-1)}. So, u′=ddx(−4)+ddx(5x−3)+ddx(7x−2)=0−15x−4−14x−3u' = \frac{d}{dx}(-4) + \frac{d}{dx}(5x^{-3}) + \frac{d}{dx}(7x^{-2}) = 0 - 15x^{-4} - 14x^{-3}.
  4. Apply Product Rule: Now, let's find the derivative of vv with respect to xx. We have v=(−2−5x−1)v=(-2-5x^{-1}). Again, the derivative of a constant is 00, and the derivative of xnx^n with respect to xx is n⋅x(n−1)n\cdot x^{(n-1)}. So, v′=ddx(−2)−ddx(5x−1)=0+5x−2v' = \frac{d}{dx}(-2) - \frac{d}{dx}(5x^{-1}) = 0 + 5x^{-2}.
  5. Simplify Expression: Now that we have u′u' and v′v', we can apply the product rule. The derivative of yy with respect to xx is given by dydx=u′v+uv′\frac{dy}{dx} = u'v + uv'. Substituting the derivatives we found, we get dydx=(−15x−4−14x−3)(−2−5x−1)+(−4+5x−3+7x−2)(5x−2)\frac{dy}{dx} = (-15x^{-4} - 14x^{-3})(-2-5x^{-1}) + (-4+5x^{-3}+7x^{-2})(5x^{-2}).
  6. Distribute u′u' with vv: We need to simplify the expression for dydx\frac{dy}{dx}. First, let's distribute u′u' with vv: (−15x−4−14x−3)(−2)+(−15x−4−14x−3)(−5x−1)(-15x^{-4} - 14x^{-3})(-2) + (-15x^{-4} - 14x^{-3})(-5x^{-1}). This simplifies to 30x−4+28x−3+75x−5+70x−430x^{-4} + 28x^{-3} + 75x^{-5} + 70x^{-4}.
  7. Distribute uu with v′v': Next, let's distribute uu with v′v': (−4)(5x−2)+(5x−3)(5x−2)+(7x−2)(5x−2)(-4)(5x^{-2}) + (5x^{-3})(5x^{-2}) + (7x^{-2})(5x^{-2}). This simplifies to −20x−2+25x−5+35x−4-20x^{-2} + 25x^{-5} + 35x^{-4}.
  8. Combine Like Terms: Now, we combine like terms to get the final expression for (dy)/(dx)(dy)/(dx). We have 30x−4+28x−3+75x−5+70x−4−20x−2+25x−5+35x−430x^{-4} + 28x^{-3} + 75x^{-5} + 70x^{-4} - 20x^{-2} + 25x^{-5} + 35x^{-4}. Combining like terms, we get (dy)/(dx)=100x−4+28x−3+100x−5−20x−2(dy)/(dx) = 100x^{-4} + 28x^{-3} + 100x^{-5} - 20x^{-2}.
  9. Final Expression: Finally, we can write the derivative in a more simplified form by factoring out the common xx term if desired. However, the expression 100x−4+28x−3+100x−5−20x−2100x^{-4} + 28x^{-3} + 100x^{-5} - 20x^{-2} is already simplified and represents the derivative of the given function with respect to xx.

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