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

Problem

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

Solution

1. Identify the function as a reciprocal or a composition of functions, which can be written as [ln(x)](−1)

2. Apply the power rule and the chain rule to differentiate the outer function.

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

5. Simplify the expression by moving the negative exponent to the denominator.

−1/(x*[ln(x)]2)

Final Answer

d()/d(x)1/ln(x)=−1/(x*ln2(x))

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