Find the Derivative - d/dx square root of x natural log of x

Problem

(d(√(,x))*ln(x))/d(x)

Solution

1. Identify the function as a product of two terms, u=√(,x) and v=ln(x) which requires the product rule.

2. Apply the product rule formula, which states that (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the individual components: d(√(,x))/d(x)=1/(2√(,x)) and d(ln(x))/d(x)=1/x

4. Substitute these derivatives back into the product rule expression.

5. Simplify the resulting terms by combining powers of x

6. Factor out common terms to reach the final simplified form.

Final Answer

(d(√(,x))*ln(x))/d(x)=(ln(x)+2)/(2√(,x))

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