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Let f(x)=cos(x)x^(-2). f^(')(x)=

Let f(x)=cos⁡(x)x−2 f(x)=\cos (x) x^{-2} .\newlinef′(x)= f^{\prime}(x)=

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Full solution

Q. Let f(x)=cos⁡(x)x−2 f(x)=\cos (x) x^{-2} .\newlinef′(x)= f^{\prime}(x)=
  1. Identify components: Identify the function components to apply the product rule: f(x)=g(x)h(x)f(x) = g(x)h(x), where g(x)=cos⁡(x)g(x) = \cos(x) and h(x)=x−2h(x) = x^{-2}.
  2. Apply product rule: Apply the product rule: f′(x)=g′(x)h(x)+g(x)h′(x)f'(x) = g'(x)h(x) + g(x)h'(x).
  3. Calculate g′(x)g'(x): Calculate g′(x)g'(x): g′(x)=−sin⁡(x)g'(x) = -\sin(x).
  4. Calculate h′(x)h'(x): Calculate h′(x)h'(x): h′(x)=−2x−3h'(x) = -2x^{-3}.
  5. Plug in values: Plug in g′(x)g'(x) and h′(x)h'(x) into the product rule: f′(x)=(−sin⁡(x))(x−2)+(cos⁡(x))(−2x−3)f'(x) = (-\sin(x))(x^{-2}) + (\cos(x))(-2x^{-3}).
  6. Simplify expression: Simplify the expression: f′(x)=−sin⁡(x)x2−2cos⁡(x)x3f'(x) = -\frac{\sin(x)}{x^2} - \frac{2\cos(x)}{x^3}.
  7. Combine terms: Combine the terms: f′(x)=−sin⁡(x)x−2cos⁡(x)x3f'(x) = \frac{-\sin(x)x - 2\cos(x)}{x^3}.

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