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

Problem

d(√(9,ln(x)))/d(x)

Solution

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

√(9,ln(x))=(ln(x))1/9

2. Apply the chain rule, which states that the derivative of ƒ*(g(x)) is ƒ′*(g(x))⋅g(x)′

d(ln(x))/d(x)=1/9*(ln(x))(1/9−1)⋅d(ln(x))/d(x)

3. Subtract the exponents and differentiate the inner function ln(x)

1/9*(ln(x))(−8/9)⋅1/x

4. Simplify the expression by combining the terms into a single fraction.

1/(9*x*(ln(x))8/9)

Final Answer

d(√(9,ln(x)))/d(x)=1/(9*x√(9,ln8(x)))

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