Question

In Exercises \(1-8,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\cos ^{-1}(1 / x)$$

Short answer

The derivative of \(y = \cos^{-1}(1/x)\) is \( dy/dx = \frac{1}{x\sqrt{x^2 - 1}} \)

Step-by-step solution

Identify the Outer and Inner Function

In the given function \(y = \cos^{-1}(1/x)\), the outer function is \( \cos^{-1}(v) \), and the inner function is \(v = 1/x \)

Differentiate the Inner Function

The derivative of the inner function \(1/x\) with respect to \(x\) comes out as \(-1/x^2\)

Differentiate the Outer Function

The derivative of the outer function \( \cos^{-1}(v) \) with respect to \(v\) is \(-1/\sqrt{1-v^2}\)

Use Chain Rule

According to the chain rule, the derivative of the composite function \( \cos^{-1}(1/x) \) is the derivative of the outer function times the derivative of the inner function. Substituting from steps 2 and 3 we get \( dy/dx = -1/\left(\sqrt{1-(1/x)^2}\right) \) times \(-1/x^2\)

Simplify

Multiply the two derivatives found from the chain rule to find the solution. This simplifies to \( dy/dx = \frac{1}{x\sqrt{x^2 - 1}} \)