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

Problem

d()/d(x)*(1/x−ln(x))

Solution

1. Identify the expression as a difference of two terms and apply the sum/difference rule for derivatives.

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

2. Rewrite the first term using a negative exponent to prepare for the power rule.

1/x=x(−1)

3. Apply the power rule to the first term, which states that d(xn)/d(x)=n*x(n−1)

d(x(−1))/d(x)=−1*x(−2)

4. Apply the derivative rule for the natural logarithm, which states that d(ln(x))/d(x)=1/x

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

5. Combine the results and simplify the negative exponent back into a fraction.

−x(−2)−1/x=−1/(x2)−1/x

Final Answer

d()/d(x)*(1/x−ln(x))=−1/(x2)−1/x

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