Find the Derivative - d/dx y=(1-x^-1)^-1

Problem

d()/d(x)*(1−x^(−1))^(−1)

Solution

1. Identify the outer and inner functions to apply the chain rule. The outer function is u^(−1) and the inner function is u=1−x^(−1)

2. Apply the power rule to the outer function. The derivative of u^(−1) is −1⋅u^(−2)

3. Differentiate the inner function 1−x^(−1) The derivative of 1 is 0 and the derivative of −x^(−1) is −(−1)*x^(−2) which simplifies to x^(−2)

4. Combine the results using the chain rule formula d(y)/d(x)=d(y)/d(u)⋅d(u)/d(x)

d(1−x^(−1))/d(x)=−1*(1−x^(−1))^(−2)⋅(x^(−2))

5. Simplify the expression by multiplying the terms.

d(1−x^(−1))/d(x)=−x^(−2)*(1−x^(−1))^(−2)

6. Rewrite using positive exponents if desired, though the power form is mathematically complete.

d(1−x^(−1))/d(x)=−1/(x^2*(1−1/x)^2)

7. Further simplify the denominator by distributing x^2 into the squared term. Since x^2*(1−1/x)^2=(x*(1−1/x))^2=(x−1)^2

Final Answer

d(1−x^(−1))/d(x)=−(x−1)^(−2)

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