Find the Derivative - d/dx f(x)=x square root of 2x-3

Problem

d()/d(x)*x√(,2*x−3)

Solution

1. Identify the function as a product of two terms, u=x and v=√(,2*x−3) which requires the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

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

d(x)/d(x)=1

3. Differentiate the second term v=√(,2*x−3) using the chain rule. Rewrite the square root as an exponent: (2*x−3)(1/2)

d(2*x−3)/d(x)=1/2*(2*x−3)(−1/2)⋅2

4. Simplify the derivative of the second term.

d(√(,2*x−3))/d(x)=1/√(,2*x−3)

5. Apply the product rule by combining the components.

(d(x)√(,2*x−3))/d(x)=x⋅1/√(,2*x−3)+√(,2*x−3)⋅1

6. Find a common denominator to combine the terms. The common denominator is √(,2*x−3)

(d(x)√(,2*x−3))/d(x)=x/√(,2*x−3)+(2*x−3)/√(,2*x−3)

7. Combine the numerators over the common denominator.

(d(x)√(,2*x−3))/d(x)=(3*x−3)/√(,2*x−3)

Final Answer

(d(x)√(,2*x−3))/d(x)=(3*x−3)/√(,2*x−3)

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