Find the Derivative - d/dx 1+1/x

Problem

d()/d(x)*(1+1/x)

Solution

1. Apply the sum rule for derivatives, which states that the derivative of a sum is the sum of the derivatives.

d()/d(x)*(1+1/x)=d(1)/d(x)+d()/d(x)1/x

2. Differentiate the constant term. The derivative of any constant is zero.

d(1)/d(x)=0

3. Rewrite the fraction using a negative exponent to prepare for the power rule.

1/x=x^(−1)

4. Apply the power rule to the term x^(−1) The power rule states that d(x^n)/d(x)=n*x^(n−1)

d(x^(−1))/d(x)=−1*x^(−2)

5. Simplify the expression by converting the negative exponent back into a fraction.

−1*x^(−2)=−1/(x^2)

6. Combine the results from the previous steps.

0+(−1/(x^2))=−1/(x^2)

Final Answer

d()/d(x)*(1+1/x)=−1/(x^2)

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