Problem 49 - Thomas Calculus

Question

In Exercises \(49-70\) , find the derivative of \(y\) with respect to the appropriate variable. $$ y=\cos ^{-1}\left(x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative of \(y\) is \(-\frac{2x}{\sqrt{1-x^4}}\)."

Step 1: Recognize the Formula for the Derivative of Inverse Cosine

The derivative of the inverse cosine function, \(\cos^{-1}(u)\), with respect to \(u\) is \(-\frac{1}{\sqrt{1-u^2}}\). This will be used once we apply the chain rule to the function. Here, \(u = x^2\).

Step 2: Apply the Chain Rule

The function \(y = \cos^{-1}(x^2)\) is comprised of an outer function, \(\cos^{-1}(u)\), and an inner function, \(u = x^2\). By the chain rule, the derivative \(\frac{dy}{dx}\) is the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of the inner function.

Step 3: Derivative of the Outer Function

Using \(u = x^2\), the derivative of the outer function \(\cos^{-1}(u)\) with respect to \(u\) is \(-\frac{1}{\sqrt{1-u^2}}\). Substitute \(u = x^2\) into this formula: \(-\frac{1}{\sqrt{1-(x^2)^2}} = -\frac{1}{\sqrt{1-x^4}}\).

Step 4: Derivative of the Inner Function

Compute the derivative of the inner function \(u = x^2\) with respect to \(x\), which is \(\frac{du}{dx} = 2x\).

Step 5: Combine Using the Chain Rule

Using the results from Steps 3 and 4, apply the chain rule: \(\frac{dy}{dx} = -\frac{1}{\sqrt{1-x^4}} \times 2x = -\frac{2x}{\sqrt{1-x^4}}\). This gives the derivative of \(y\) with respect to \(x\).

Step 6: Verify the Result

Ensure each step has been correctly applied, specifically checking the derivative rules and chain rule application. The chain rule combines the two derivatives into \(-\frac{2x}{\sqrt{1-x^4}}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse trigonometric operations, essentially answering the question: what angle creates this trigonometric result? They play a critical role in calculus when dealing with derivatives and integrals because they provide unique solutions for angles given certain values. The inverse cosine function, denoted as \( \cos^{-1}(x) \), is particularly important. It specifies the angle whose cosine equals \(x\). When differentiating inverse trigonometric functions, formulas help guide us. For \( \cos^{-1}(x) \), the derivative formula is

• \( \frac{d}{dx} [\cos^{-1}(x)] = -\frac{1}{\sqrt{1-x^2}} \)

This formula only works for values of \(x\) that keep \( \sqrt{1-x^2} \) real and non-negative, precisely when \(-1 \leq x \leq 1\). When composite functions involve inverse trigonometric functions, they often require the usage of additional differentiation techniques, like the chain rule.

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. A composite function arises when one function is nested within another, like \( f(g(x)) \). To differentiate such a composite function, the chain rule states:

• \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \)

This tells us to differentiate the outer function with respect to its inner function (\(u\)) and multiply by the derivative of the inner function with respect to \(x\).Consider the exercise: \( y = \cos^{-1}(x^2) \). Here \( y = \cos^{-1}(u) \) is the outer function where \( u = x^2 \) is the inner function. The derivative of \( \cos^{-1}(u) \) is \( -\frac{1}{\sqrt{1-u^2}} \), and \( \frac{du}{dx} \) is \(2x\). So, applying the chain rule, you combine these derivatives to get the complete derivative:

• \( \frac{dy}{dx} = -\frac{1}{\sqrt{1-x^4}} \times 2x = -\frac{2x}{\sqrt{1-x^4}} \)

This shows how the chain rule simplifies tackling nested functions.

Differentiation Techniques

Differentiation is the process of finding the rate at which a function changes at any point, known as its derivative. Several techniques can simplify finding derivatives, especially for more complex functions. Key techniques include:

• **Basic Rules**: Power rule, product rule, and quotient rule help with direct functions of \(x\).
• **Chain Rule**: Used, as explained, for composite functions with nested layers.
• **Inverse Function Differentiation**: Specific formulas exist for inverse trigonometric functions, as seen in \( \cos^{-1}(x) \) differentiation.

These techniques are applied in sequence or combination to resolve the derivatives of more advanced functions.In the given exercise, combining these methods brings efficiency. Initially, recognizing the inverse function and applying its specific derivative formula is crucial. Next, using the chain rule links outer and inner derivatives effectively. This ordered approach ensures that each component of a composite function is tackled precisely, leading to accurate derivative computation.