Find the 2nd Derivative f(x) = natural log of x

Problem

ƒ(x)=ln(x)

Solution

1. Identify the function to be differentiated, which is ƒ(x)=ln(x)

2. Find the first derivative by applying the power rule for logarithms, where d(ln(x))/d(x)=1/x

3. Rewrite the first derivative as a power to prepare for the next differentiation: ƒ(x)^′=x^(−1)

4. Find the second derivative by applying the power rule d(x^n)/d(x)=n*x^(n−1) to x^(−1)

5. Simplify the resulting expression ƒ(x)^″=−1*x^(−2) by moving the variable back to the denominator.

Final Answer

d^2(ln(x))/(d(x)^2)=−1/(x^2)

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