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Find the derivative of f(x). f(x)=sqrt(x-1)

Find the derivative of f(x) f(x) .\newlinef(x)=x−1 f(x)=\sqrt{x-1}

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Full solution

Q. Find the derivative of f(x) f(x) .\newlinef(x)=x−1 f(x)=\sqrt{x-1}
  1. Apply Chain Rule: To find the derivative of f(x)=x−1f(x) = \sqrt{x-1}, we need to apply the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
  2. Identify Outer Function: The outer function is the square root function, which can be written as x-1)^{\frac{1}{2}}\. The derivative of \$x^{\frac{1}{2}} with respect to xx is 12x−12\frac{1}{2}x^{-\frac{1}{2}}.
  3. Identify Inner Function: The inner function is (x−1)(x-1). The derivative of (x−1)(x-1) with respect to xx is 11, since the derivative of a constant is 00 and the derivative of xx is 11.
  4. Apply Chain Rule Again: Applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. This gives us:\newlinef′(x)=(12)(x−1)−12×1f'(x) = (\frac{1}{2})(x-1)^{-\frac{1}{2}} \times 1
  5. Simplify Final Derivative: Simplify the expression to get the final derivative: f′(x)=12x−1f'(x) = \frac{1}{2\sqrt{x-1}}

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