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

Problem

d(ln(√(,x)))/d(x)

Solution

1. Rewrite the expression using the properties of exponents to simplify the square root.

√(,x)=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. Identify the constant multiple rule for derivatives, which allows the constant factor to be moved outside the derivative.

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

4. Differentiate the natural log function using the rule d(ln(x))/d(x)=1/x

1/2⋅1/x=1/(2*x)

Final Answer

d(ln(√(,x)))/d(x)=1/(2*x)

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