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+u2)

4. Differentiate the inner function using the power rule: d(x(−1))/d(x)=−1*x(−2)=−1/(x2)

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/(x2))

6. Simplify the expression by distributing x2 into the denominator.

d(arctan(1/x))/d(x)=−1/(x2*(1+1/(x2)))

d(arctan(1/x))/d(x)=−1/(x2+1)

Final Answer

d(arctan(1/x))/d(x)=−1/(x2+1)

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