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f(x)=xx−1f(x) = \frac{\sqrt{x}}{x-1}

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Find derivatives using the quotient rule Iright chevron icon

Full solution

Q. f(x)=xx−1f(x) = \frac{\sqrt{x}}{x-1}
  1. Identify function components: Identify the function components: the numerator is x\sqrt{x}, which is x(1/2)x^{(1/2)}, and the denominator is (x−1)(x-1).
  2. Apply quotient rule: Apply the quotient rule: (v⋅(dudx)−u⋅(dvdx))/(v2)(v \cdot (\frac{du}{dx}) - u \cdot (\frac{dv}{dx})) / (v^{2}), where u=x12u = x^{\frac{1}{2}} and v=x−1v = x - 1.
  3. Differentiate numerator: Differentiate the numerator: (dudx)=ddx(x12)=(12)x−12(\frac{du}{dx}) = \frac{d}{dx}(x^{\frac{1}{2}}) = (\frac{1}{2})x^{-\frac{1}{2}}.
  4. Differentiate denominator: Differentiate the denominator: (dvdx)=ddx(x−1)=1(\frac{dv}{dx}) = \frac{d}{dx}(x - 1) = 1.
  5. Plug in derivatives: Plug in the derivatives into the quotient rule: f′(x)=(x−1)⋅(12x−12)−x12⋅1(x−1)2f'(x) = \frac{(x - 1) \cdot (\frac{1}{2}x^{-\frac{1}{2}}) - x^{\frac{1}{2}} \cdot 1}{(x - 1)^2}.
  6. Simplify expression: Simplify the expression: f′(x)=(12x12−x12)(x−1)2f'(x) = \frac{(\frac{1}{2}x^{\frac{1}{2}} - x^{\frac{1}{2}})}{(x - 1)^2}.

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