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

Problem

d(ln(√(5,x)))/d(x)

Solution

1. Rewrite the fifth root using a fractional exponent to make the expression easier to differentiate.

√(5,x)=x(1/5)

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

ln(x(1/5))=1/5*ln(x)

3. Identify the constant multiple rule for derivatives, which allows the constant 1/5 to be moved outside the derivative.

d()/d(x)1/5*ln(x)=1/5d(ln(x))/d(x)

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

1/5⋅1/x=1/(5*x)

Final Answer

d(ln(√(5,x)))/d(x)=1/(5*x)

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