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

Problem

d(√(3,ln(x)))/d(x)

Solution

1. Rewrite the expression using a fractional exponent to prepare for differentiation.

√(3,ln(x))=(ln(x))(1/3)

2. Apply the power rule and the chain rule, where the outer function is u(1/3) and the inner function is ln(x)

d(ln(x))/d(x)=1/3*(ln(x))(−2/3)⋅d(ln(x))/d(x)

3. Differentiate the inner function ln(x) with respect to x

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

4. Substitute the derivative of the inner function back into the expression.

1/3*(ln(x))(−2/3)⋅1/x

5. Simplify the expression by moving the negative exponent to the denominator and combining terms.

1/(3*x*(ln(x))(2/3))

Final Answer

d(√(3,ln(x)))/d(x)=1/(3*x√(3,(ln(x))2))

Want more problems? Check here!