Find the Derivative - d/dx f(x) = square root of x-1

Problem

d()/d(x)√(,x−1)

Solution

1. Rewrite the square root expression using a fractional exponent to prepare for the power rule.

√(,x−1)=(x−1)(1/2)

2. Apply the chain rule, which states that the derivative of un is n*u(n−1)⋅d(u)/d(x)

d(x−1)/d(x)=1/2*(x−1)(1/2−1)⋅d(x−1)/d(x)

3. Differentiate the inner function x−1 with respect to x

d(x−1)/d(x)=1

4. Simplify the expression by combining the terms and moving the negative exponent to the denominator.

1/2*(x−1)(−1/2)⋅1=1/(2*(x−1)(1/2))

5. Convert the fractional exponent back into radical form.

1/(2√(,x−1))

Final Answer

d(√(,x−1))/d(x)=1/(2√(,x−1))

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