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/(x2*(1−1/x)2)

7. Further simplify the denominator by distributing x2 into the squared term. Since x2*(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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