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Let f(x)=x^(-1). f^(')(x)=

Let f(x)=x−1 f(x)=x^{-1} .\newlinef′(x)= f^{\prime}(x)=

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Full solution

Q. Let f(x)=x−1 f(x)=x^{-1} .\newlinef′(x)= f^{\prime}(x)=
  1. Identify Function: Identify the function to differentiate.\newlineWe are given the function f(x)=x−1f(x) = x^{-1}, which is a power function.
  2. Apply Power Rule: Apply the power rule for differentiation. The power rule states that if f(x)=xnf(x) = x^n, then f′(x)=n⋅x(n−1)f^{\prime}(x) = n \cdot x^{(n-1)}. Here, n=−1n = -1.
  3. Differentiate Function: Differentiate the function using the power rule.\newlinef′(x)=(−1)×x(−1−1)f'(x) = (-1) \times x^{(-1 - 1)}\newlinef′(x)=(−1)×x−2f'(x) = (-1) \times x^{-2}
  4. Simplify Expression: Simplify the expression.\newlinef′(x)=−x−2f^{\prime}(x) = -x^{-2}\newlineThis is the derivative of the function f(x)=x−1f(x) = x^{-1}.

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