Find the Derivative - d/dx arctan(1/x)

Problem

d(arctan(1/x))/d(x)

Solution

1. Identify the outer function as ƒ(u)=arctan(u) and the inner function as u=1/x=x^(−1)

2. Apply the chain rule, which states that d(ƒ(u))/d(x)=d(ƒ)/d(u)⋅d(u)/d(x)

3. Differentiate the outer function with respect to u using the rule d(arctan(u))/d(u)=1/(1+u^2)

4. Differentiate the inner function using the power rule: d(x^(−1))/d(x)=−1*x^(−2)=−1/(x^2)

5. Substitute u=1/x back into the derivative of the outer function and multiply by the derivative of the inner function.

d(arctan(1/x))/d(x)=1/(1+(1/x)^2)⋅(−1/(x^2))

6. Simplify the expression by distributing x^2 into the denominator.

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

d(arctan(1/x))/d(x)=−1/(x^2+1)

Final Answer

d(arctan(1/x))/d(x)=−1/(x^2+1)

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