Question

Compute the derivative of the given function. $$f(t)=\sec ^{-1}(2 t)$$

Short answer

The derivative is \(\frac{1}{|t| \sqrt{4t^2-1}}\).

Step-by-step solution

Understand the Function

The function given is \(f(t) = \sec^{-1}(2t)\), the inverse secant function. Our goal is to find its derivative.

Recall the Derivative of Inverse Secant

The derivative of \(\sec^{-1}(u)\) with respect to \(u\) is \(\frac{1}{|u| \sqrt{u^2-1}}\). Here, \(u = 2t\).

Apply the Chain Rule

Using the chain rule, the derivative of \(f(t)\) is: \[ \frac{d}{dt} f(t) = \frac{d}{dt} \sec^{-1}(2t) = \frac{d}{d(2t)} \sec^{-1}(2t) \times \frac{d}{dt}(2t).\]

Derivative of the Inside Function

The derivative of \(2t\) with respect to \(t\) is 2.

Combine the Derivatives

Substitute \(u = 2t\) and \(\frac{du}{dt} = 2\) into the derivative formula from Step 2: \[ \frac{d}{dt} \sec^{-1}(2t) = \frac{1}{|2t| \sqrt{(2t)^2-1}} \times 2.\]

Simplify the Expression

Simplify the expression to find the derivative:\[ \frac{2}{|2t| \sqrt{4t^2-1}} = \frac{1}{|t| \sqrt{4t^2-1}}.\]Therefore, the derivative of \(f(t) = \sec^{-1}(2t)\) is \(\frac{1}{|t| \sqrt{4t^2-1}}\).

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the basic trigonometric functions. They allow us to work backwards to find the angles when given a trigonometric ratio. For example, the inverse secant, denoted as \(\sec^{-1}(x)\), answers the question: "What angle has a secant value of \(x\)?"
These functions are crucial in calculus when we're faced with scenarios that involve reversing trigonometric operations. Their derivatives might seem a little tricky at first, but they follow a specific pattern.
Unlike ordinary functions, the derivatives of inverse trigonometric functions introduce terms involving absolute values and square roots. This is because of their connection to the unit circle and the constraints placed on these functions to be properly defined and 'one-to-one'. Understanding these derivatives is essential for solving problems that involve more complex expressions of angles.

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. When you have a function nested within another function, the chain rule helps us find the derivative of the outer function while considering the derivative of the inner function.
Here's how it works in simple terms:

• First, differentiate the outer function while leaving the inner function unchanged.
• Second, multiply this result by the derivative of the inner function.

In the provided exercise, \(f(t) = \sec^{-1}(2t)\) is a composite function where \(\sec^{-1}\) is the outer function and \(2t\) is the inner function. When applying the chain rule here, we first differentiate \(\sec^{-1}(2t)\) with respect to \(2t\), then multiply the entire derivative by the derivative of \(2t\) with respect to \(t\), which is 2. This combination of steps using the chain rule yields the correct derivative expression for the function.

Derivative of Inverse Secant

When differentiating the inverse secant function \(\sec^{-1}(u)\), it's important to remember the derivative formula. The derivative of \(\sec^{-1}(u)\) with respect to \(u\) is \(\frac{1}{|u| \sqrt{u^2-1}}\). This formula captures the unique behavior of the inverse secant, which arises because of the properties of the secant function and the geometry of a circle.
In our problem, the function involves \(\sec^{-1}(2t)\). By recognizing \(u\) as \(2t\), and applying the derivative formula, we find it necessary to transform the expression using the chain rule to account for the inner derivative \(\frac{d}{dt}(2t)\), which is 2.
Substituting back into our equation after applying the chain rule gives us a complete expression: \[ \frac{d}{dt} \sec^{-1}(2t) = \frac{2}{|2t| \sqrt{4t^2-1}}.\] Further simplification leads us to the final derivative: \(\frac{1}{|t| \sqrt{4t^2-1}}\). This is the neat, clear expression that represents the rate of change of the inverse secant with a linear function inside.