Find the Derivative - d/dx natural log of (x)^(1/3)

Problem

d(ln(x(1/3)))/d(x)

Solution

1. Apply the logarithm power rule to simplify the expression before differentiating. The rule ln(ab)=b*ln(a) allows the exponent to be moved in front of the natural log.

ln(x(1/3))=1/3*ln(x)

2. Apply the constant multiple rule for derivatives. This rule states that (d(c)⋅ƒ(x))/d(x)=c⋅d(ƒ(x))/d(x)

d()/d(x)1/3*ln(x)=1/3d(ln(x))/d(x)

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

1/3⋅1/x

4. Multiply the fractions to find the final derivative.

1/(3*x)

Final Answer

d(ln(x(1/3)))/d(x)=1/(3*x)

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