Find the Derivative - d/dx x square root of x-1

Problem

d()/d(x)*x√(,x−1)

Solution

1. Identify the function as a product of two terms, u=x and v=√(,x−1) which requires the product rule (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Rewrite the square root as a power to prepare for differentiation.

√(,x−1)=(x−1)(1/2)

3. Differentiate the first term u=x with respect to x

d(x)/d(x)=1

4. Differentiate the second term v=(x−1)(1/2) using the power rule and the chain rule.

d(x−1)/d(x)=1/2*(x−1)(−1/2)⋅1

d(x−1)/d(x)=1/(2√(,x−1))

5. Apply the product rule formula by substituting the derivatives found in the previous steps.

(d(x)√(,x−1))/d(x)=x⋅1/(2√(,x−1))+√(,x−1)⋅1

6. Simplify the expression by finding a common denominator to combine the terms.

(d(x)√(,x−1))/d(x)=x/(2√(,x−1))+(2*(x−1))/(2√(,x−1))

(d(x)√(,x−1))/d(x)=(x+2*x−2)/(2√(,x−1))

(d(x)√(,x−1))/d(x)=(3*x−2)/(2√(,x−1))

Final Answer

(d(x)√(,x−1))/d(x)=(3*x−2)/(2√(,x−1))

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