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

Problem

d(ln(√(,x)))/d(x)

Solution

1. Rewrite the expression using the properties of exponents by expressing the square root as a power of one-half.

ln(√(,x))=ln(x(1/2))

2. Apply the logarithm power rule, ln(ab)=b*ln(a) to move the exponent in front of the natural log.

ln(x(1/2))=1/2*ln(x)

3. Differentiate the simplified expression using the constant multiple rule and the derivative of the natural log, d(ln(x))/d(x)=1/x

d()/d(x)1/2*ln(x)=1/2⋅1/x

4. Simplify the resulting fraction by multiplying the terms.

1/(2*x)

Final Answer

d(ln(√(,x)))/d(x)=1/(2*x)

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