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

Problem

d(ln(√(3,x)))/d(x)

Solution

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

√(3,x)=x(1/3)

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

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

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

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

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

1/3⋅1/x=1/(3*x)

Final Answer

d(ln(√(3,x)))/d(x)=1/(3*x)

Want more problems? Check here!