Question

Compute the derivative of the given function. $$f(x)=\sec ^{-1}(1 / x)$$

Short answer

The derivative is \( -\frac{1}{x \sqrt{1 - x^2}} \).

Step-by-step solution

Identify the Function Type

The function given is the inverse secant function applied to the reciprocal of x, that is, \( f(x) = \sec^{-1}\left(\frac{1}{x}\right) \). This means we will need the derivative formula for the inverse secant function.

Recall the Derivative of Inverse Secant Function

The derivative of \( \sec^{-1}(u) \) with respect to \( u \) is \( \frac{1}{|u| \sqrt{u^2 - 1}} \). In this problem, \( u = \frac{1}{x} \). We must also consider the chain rule for differentiation.

Differentiate the Inside Function \(u = \frac{1}{x}\)

Differentiate \( u = \frac{1}{x} \) with respect to \( x \), which gives \( \frac{du}{dx} = -\frac{1}{x^2} \).

Apply the Chain Rule

The derivative of \( f(x) = \sec^{-1}\left(\frac{1}{x}\right) \) using the chain rule is \( \frac{df}{dx} = \frac{1}{|1/x| \sqrt{(1/x)^2 - 1}} \cdot \frac{du}{dx} \). Substituting \( \frac{du}{dx} = -\frac{1}{x^2} \) from Step 3.

Simplify the Derivative Expression

First compute \( |1/x| \sqrt{(1/x)^2 - 1} = \frac{1}{|x|} \sqrt{\frac{1}{x^2} - 1} = \frac{1}{|x|} \sqrt{\frac{1 - x^2}{x^2}} \). The derivative becomes \( \frac{df}{dx} = -\frac{1}{x^2} \cdot \frac{x}{\sqrt{1 - x^2}} = -\frac{1}{x \sqrt{1 - x^2}} \).

Final Step: Conclusion

Therefore, the derivative of \( f(x) = \sec^{-1}\left(\frac{1}{x}\right) \) is \( -\frac{1}{x \sqrt{1 - x^2}} \).

Key concepts

Derivative

The derivative of a function measures how the function's output changes as its input changes. In calculus, the derivative is a fundamental concept used to understand the behavior of functions. It's particularly important in fields like physics, where it's used to determine rates of change such as speed or acceleration. For every function you can think of, there exists a set of rules and methods to compute its derivative. This often involves recalling specific formulas or applying various techniques, depending on the type of function you're dealing with.

Inverse Functions

Inverse functions are functions that reverse the effect of the original function. For example, if a function \( f \) takes \( x \) to \( y \), then its inverse, denoted as \( f^{-1} \), takes \( y \) back to \( x \). Working with inverse functions requires understanding that they essentially "undo" what the initial function does. Not every function has an inverse, but when they do, they must pass the "horizontal line test," meaning each horizontal line should intersect the curve of the function at most once. The inverse secant function \( \sec^{-1}(x) \), specifically, is often discussed with trigonometric functions and is pivotal in scenarios where you need to reverse the secant function.

Chain Rule

The chain rule is a vital method in calculus for finding the derivative of composite functions. This means computing the derivative of a function that's applied to another function, like \( f(g(x)) \). The chain rule formula states that if you have two functions \( f \) and \( g \), then the derivative of their composite is \( f'(g(x)) \cdot g'(x) \).
The key idea here is that you "chain" together the derivatives of the individual functions, multiplying them to get the derivative of the composite function. When dealing with problems involving the chain rule, always be mindful of differentiating the inner function first, followed by the outer function, using the derivative of the inner function where needed.

Inverse Secant Function

The inverse secant function \( \sec^{-1}(x) \) is one of the inverse trigonometric functions. It's especially useful when we need to solve for angles in right triangles when the secant (the reciprocal of cosine) is known. The derivative of the inverse secant is a bit more involved than that of simpler functions. In particular, for a given \( u \), the derivative \( \frac{d}{du}(\sec^{-1}(u)) \) is \( \frac{1}{|u|\sqrt{u^2 - 1}} \).
Understanding this derivative helps in cases where complex functions, such as those in physics or engineering, are involved, and you need to find how a change in one variable affects another. In contexts that involve composing transformations with trigonometric principles, the inverse secant and its derivative play a crucial role.