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

Problem

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

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is arccos(u) and the inner function is u=1/x

2. Recall the derivative formula for the inverse cosine function, which is d(arccos(u))/d(u)=−1/√(,1−u2)

3. Apply the Chain Rule by multiplying the derivative of the outer function by the derivative of the inner function.

d(y)/d(x)=−1/√(,1−(1/x)2)⋅d()/d(x)1/x

4. Differentiate the inner function 1/x which is x(−1) using the Power Rule to get −x(−2) or −1/(x2)

d(y)/d(x)=−1/√(,1−1/(x2))⋅(−1/(x2))

5. Simplify the expression by multiplying the terms and simplifying the square root in the denominator.

d(y)/d(x)=1/(x2√(,(x2−1)/(x2)))

6. Simplify further by taking the square root of x2 in the denominator, noting that √(,x2)=|x|

d(y)/d(x)=1/(x2√(,x2−1)/|x|)

7. Reduce the expression by canceling x terms, assuming x>1 or x<−1 for the domain of the original function.

d(y)/d(x)=1/(|x|√(,x2−1))

Final Answer

d()/d(x)*arccos(1/x)=1/(|x|√(,x2−1))

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