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

Problem

d(arccos(1/x))/d(x)

Solution

1. Identify the outer function as arccos(u) and the inner function as u=1/x

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

3. Differentiate the outer function using the formula d(arccos(u))/d(u)=−1/√(,1−u2)

4. Differentiate the inner function u=x(−1) using the power rule to get d(u)/d(x)=−x(−2)=−1/(x2)

5. Substitute the expressions back into the chain rule formula.

d(arccos(1/x))/d(x)=−1/√(,1−(1/x)2)⋅(−1/(x2))

6. Simplify the expression by multiplying the terms and simplifying the square root.

d(arccos(1/x))/d(x)=1/(x2√(,1−1/(x2)))

7. Distribute x into the square root (noting √(,x2)=|x| to further simplify.

d(arccos(1/x))/d(x)=1/(x2√(,(x2−1)/(x2)))

d(arccos(1/x))/d(x)=1/(x2√(,x2−1)/|x|)

d(arccos(1/x))/d(x)=1/(|x|√(,x2−1))

Final Answer

d(arccos(1/x))/d(x)=1/(|x|√(,x2−1))

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