Problem 21 - Thomas Calculus

Question

In Exercises \(21-42,\) 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 is \(-\frac{2x}{\sqrt{1-x^4}}\)."

Step 1: Identify the Function and Variable

We are given the function \(y = \cos^{-1}(x^2)\). Our task is to find the derivative \( \frac{dy}{dx} \), where \(x\) is the variable.

Step 2: Use Chain Rule

The function \(y = \cos^{-1}(u)\) has an inner function \(u = x^2\). The chain rule states that the derivative of a composite function \(y = g(u(x))\) is \( \frac{dy}{dx} = \frac{dg}{du} \cdot \frac{du}{dx} \).

Step 3: Differentiate the Inner Function

Find the derivative of the inner function \(u = x^2\) with respect to \(x\): \( \frac{du}{dx} = 2x \).

Step 4: Differentiate the Outer Function

The derivative of the outer function \(g(u) = \cos^{-1}(u)\) with respect to \(u\) is \( \frac{dg}{du} = -\frac{1}{\sqrt{1-u^2}} \).

Step 5: Combine Using Chain Rule

Multiply the derivatives from Steps 3 and 4 using the chain rule: \( \frac{dy}{dx} = -\frac{1}{\sqrt{1-(x^2)^2}} \cdot 2x \).

Step 6: Simplify the Expression

Simplify the expression for \( \frac{dy}{dx} \) to get: \( \frac{dy}{dx} = -\frac{2x}{\sqrt{1-x^4}} \).

Key Concepts

Chain Rule

The chain rule is an essential concept in calculus for finding the derivative of composite functions. It allows us to differentiate functions that are nested within other functions, breaking them down into simpler parts. To apply the chain rule, you first identify the outer and inner functions. The formula for the chain rule is:

• If a function is expressed as a composite, i.e., \( y = g(f(x)) \), then the derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = g'(f(x)) \cdot f'(x) \).

In simpler terms, you take the derivative of the outer function, multiplying it by the derivative of the inner function. In our exercise, \( y = \cos^{-1}(x^2) \), \( \cos^{-1}(u) \) is the outer function and \( u = x^2 \) is the inner function. By applying the chain rule, we combine their derivatives for the final solution.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse operations of the traditional sine, cosine, and tangent functions. They help us find angles when we know the ratios of the sides of a right triangle. The notation for inverse trigonometric functions include \( \sin^{-1}, \cos^{-1} \), and \( \tan^{-1} \).
These functions have specific derivatives, which are crucial while solving calculus problems. For instance, the derivative of \( y = \cos^{-1}(u) \) with respect to \( u \) is:

• \( \frac{d}{du} \cos^{-1}(u) = -\frac{1}{\sqrt{1-u^2}} \)

This derivative is used to simplify and solve more complex differentiation problems like the one in the original exercise, which involves a composite function with \( \cos^{-1} \). Understanding these derivatives helps navigate the differentiation of functions involving trigonometric parents.

Differentiation Techniques

Differentiation techniques cover a vast area in calculus, providing methods to find derivatives of various function types. Common techniques include basic rules like the power rule and sum rule. More sophisticated methods incorporate the chain rule, product rule, and quotient rule.

• The power rule: For \( f(x) = x^n \), the derivative \( f'(x) = nx^{n-1} \).
• The sum rule: If \( f(x) = a(x) + b(x) \), then \( f'(x) = a'(x) + b'(x) \).
• Chain rule: Essential for composite functions to differentiate nested functions.

In the given problem, differentiating the inner function \( u = x^2 \) involves the power rule, yielding \( \frac{du}{dx} = 2x \). Coupled with the outer function's derivative using the chain rule, these techniques provide a powerful approach to finding derivatives of more intricate functions.

Composite Functions

Composite functions are formed when one function is applied to the result of another function. This combined operation is represented as \( f(g(x)) \). To correctly differentiate such functions, it is crucial to identify both the inner and outer functions.

• Inner function: Typically, the function nearest to the variable being differentiated, for example, \( u = x^2 \) in this case.
• Outer function: Operates on the result of the inner function, such as \( \cos^{-1}(u) \).

The chain rule is applied to find the derivative. Composite functions can appear complex but breaking them down simplifies differentiation. In our exercise, the composite nature of \( y = \cos^{-1}(x^2) \) involved identifying \( x^2 \) as the inner function and \( \cos^{-1} \) as the outer function, using appropriate differentiation techniques.