Problem 22 - Thomas Calculus

Question

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\cos ^{-1}(1 / x) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the appropriate variable

We are given the function \( y = \cos^{-1}(1/x) \). The variable with respect to which we need to find the derivative is \( x \).

Step 2: Recall the derivative of inverse cosine

The derivative of \( \cos^{-1}(u) \) with respect to \( u \) is given by \[ \frac{d}{du}[\cos^{-1}(u)] = -\frac{1}{\sqrt{1-u^2}}. \] This formula will be used to find the derivative of our function.

Step 3: Determine the inner function and its derivative

In this function, \( u = 1/x \). The derivative of \( u \) with respect to \( x \) is \( \frac{d}{dx}(1/x) = -\frac{1}{x^2} \).

Step 4: Use the chain rule

Using the chain rule for differentiation, we combine the derivative of the outer function with the derivative of the inner function: \[ \frac{dy}{dx} = \frac{d}{du}[\cos^{-1}(u)] \cdot \frac{d}{dx}(u). \] Substitute \( u = 1/x \) and its derivative: \[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - (1/x)^2}} \cdot \left(-\frac{1}{x^2}\right). \]

Step 5: Simplify the expression

Let's simplify the expression further. We can start by simplifying \( 1 - (1/x)^2 = 1 - 1/x^2 = \frac{x^2 - 1}{x^2} \). Now simplify the main expression: \[ \frac{dy}{dx} = \frac{1}{\sqrt{\frac{x^2 - 1}{x^2}}} \cdot \frac{1}{x^2} \] \[ = \frac{x}{\sqrt{x^2 - 1}} \cdot \frac{1}{x^2}. \] Finally, we get: \[ \frac{dy}{dx} = \frac{1}{x\sqrt{x^2 - 1}}. \]

Key Concepts

Chain Rule

A crucial tool in calculus is the chain rule, especially when dealing with composite functions like our given problem, \(y = \cos^{-1}(1/x)\). In essence, the chain rule states that to differentiate a composite function, you take the derivative of the outer function and multiply it by the derivative of the inner function.

In this example, the outer function is \(\cos^{-1}(u)\) and the inner function is \(u = 1/x\). Here’s how you apply the chain rule:

• First, find the derivative of the outer function with respect to its variable \(u\).
• Then, find the derivative of the inner function with respect to its variable \(x\).
• Multiply these two derivatives together to get your result.

The chain rule greatly simplifies the process of differentiating complex functions by breaking them down into smaller and easier-to-handle parts.

Differentiation Techniques

The world of calculus provides several differentiation techniques, each suited for different types of functions. For our problem of differentiating \(y = \cos^{-1}(1/x)\), applying the chain rule is one such technique.

Conversely, using symbolic rules of differentiation or basic derivatives like the power rule, product rule, or quotient rule becomes essential in managing basic to compound functions. However, when you encounter inverse functions or nested functions like \(1/x\) inside an inverse trigonometric function, you rely heavily on specific formulas and rules like the derivative of inverse trig functions in combination with the chain rule.

• Understand and memorize the basic derivative formula for inverse trigonometric functions.
• Practice finding the derivative of inner functions separately; it helps when applying the chain rule.
• Simplify expressions stepwise to reduce errors in complex derivations.

These techniques, when used appropriately, streamline the process and increase accuracy while working through calculus problems.

Inverse Cosine Function

Inverse trigonometric functions, such as the inverse cosine function \(\cos^{-1}(u)\), are used to calculate angles given a cosine value. However, when you need to differentiate them, you need to remember their special derivatives.

For the inverse cosine function, the derivative \(\frac{d}{du}[\cos^{-1}(u)]\) is \(-\frac{1}{\sqrt{1-u^2}}\). This formula is essential since it provides a shortcut to finding slopes of graphs that involve inverse cosine, as seen in our exercise. It highlights how the variation in the cosine inverse function converges back to simplicity by providing a pattern to follow.

• Pay attention to the domain restrictions: The inverse cosine is only defined for \(u\) between \(-1\) and \(1\).
• Remember this pattern helps decode lots of seemingly difficult problems involving inverse trig functions.
• Connections between these functions and angles become significant for interpreting physical versus theoretical worlds.

Mastering the derivative of these functions streamlines calculus processes and enhances your problem-solving skills.

Calculus Exercises

When working through calculus exercises, especially in differentiation, understanding the core principles is vital. To handle exercises like differentiating \(y = \cos^{-1}(1/x)\), it's essential to keep a few strategies in mind.

Start by analyzing what type of function you are dealing with and determine the best approach for differentiation. Follow these steps:

• Identify the function type and its components. In our case, it’s a composite of a trigonometric inverse and a rational function.
• Apply appropriate rules, like the chain rule, to efficiently solve the problem.
• Double-check simplifications at each step to ensure accuracy.
• Rework similar types of problems to gain confidence and proficiency.

Practicing these techniques with step-by-step examples like in our original solution boosts understanding and retention, making future exercises easier and more intuitive to solve.