Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \begin{array}{l} \text { Use the definition of the derivative to show that }\\\ D_{x}\left(\sin x^{2}\right)=2 x \cos x^{2} \end{array} $$

Step-by-Step Solution

Answer

The derivative is shown to be \( 2x \cos(x^2) \).

Step 1: Understand the Definition of the Derivative

The definition of the derivative at a point is given by \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). We will apply this to the function \( \sin(x^2) \) to find \( D_x(\sin x^2) \).

Step 2: Set Up the Derivative Using Definition

Apply the definition to the function \( f(x) = \sin(x^2) \). We need to find \( \lim_{h \to 0} \frac{\sin((x+h)^2) - \sin(x^2)}{h} \).

Step 3: Use the Angle Sum for Sine

Recall that \( \sin(A+B) = \sin A \cos B + \cos A \sin B \). Therefore, we can express \( \sin((x+h)^2) \) using \( A = x^2 \) and \( B = 2xh + h^2 \).

Step 4: Simplify the Expression

Calculate using the angle sum formula: \( \sin((x+h)^2) = \sin(x^2 + 2xh + h^2) = \sin(x^2)\cos(2xh + h^2) + \cos(x^2)\sin(2xh + h^2) \).

Step 5: Approximate Small Angles

For small \( h \), \( \cos(2xh + h^2) \approx 1 \) and \( \sin(2xh + h^2) \approx 2xh + h^2 \). Hence the expression becomes \( \sin(x^2) + \cos(x^2)(2xh + h^2) \).

Step 6: Apply the Limit

Substitute into the difference quotient: \[ \frac{\sin((x+h)^2) - \sin(x^2)}{h} \approx \frac{\sin(x^2) + 2xh\cos(x^2) + h^2\cos(x^2) - \sin(x^2)}{h} \]. The \( \sin(x^2) \) terms cancel, and we simplify to \( 2x\cos(x^2) + h\cos(x^2) \).

Step 7: Complete the Derivative Calculation

Take the limit as \( h \to 0 \): \[ \lim_{h \to 0} (2x \cos(x^2) + h \cos(x^2)) = 2x \cos(x^2) \].

Key Concepts

Definition of Derivative

The derivative of a function at a given point describes the rate of change, or the slope, of the function at that particular point. It acts as the fundamental tool for understanding how functions change. In our exercise, we began by using this basic definition of a derivative:

• The derivative of a function \( f(x) \) at any point \( x \) is represented as \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).
• This formula essentially calculates the function's instantaneous rate of change.

In applying this to \( \sin(x^2) \), we recognize that substituting \( f(x) = \sin(x^2) \) allows us to explore how the sine function evolves with small modifications in the input, \( x^2 \). By determining how slight changes in the input affect the sine function, you're essentially tracing the tangent line to the curve, constructing the derivative from the ground up. Understanding this core concept of limit-based construction underlies the entire process of differentiation.

Angle Sum Formula

The angle sum formula is a crucial identity in trigonometry that assists in breaking down complex sinusoidal expressions. Specifically, for sine, it can be expressed as:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

In the context of differentiating \( \sin(x^2) \), we are tasked with finding \( \sin((x+h)^2) \). Here, the formula is used to rewrite this as a sum of angles, \( A = x^2 \) and \( B = 2xh + h^2 \). This decomposition aids tremendously by allowing each part to be evaluated easier:

• First, rewrite \( \sin((x+h)^2) \) using the identity: \( \sin(x^2) \cos(2xh + h^2) + \cos(x^2) \sin(2xh + h^2) \).
• This step is integral in dissecting the sine of a squared sum into manageable components that are easier to differentiate.

By leveraging this formula, you're breaking down a potentially overwhelming expression into digestible parts, each of which can be separately analyzed and approximated, as seen in the exercise with the aid of small angle approximations.

Small Angle Approximation

When dealing with derivatives and limits involving small quantities, small angle approximations become exceptionally useful. These approximations provide a simplified way to approximate the behavior of trigonometric functions:

• For small angles \( \theta \), \( \cos(\theta) \approx 1 \) and \( \sin(\theta) \approx \theta \).

Applying this to our context, for the small quantity \( 2xh + h^2 \), we have:

• \( \cos(2xh + h^2) \approx 1 \)
• \( \sin(2xh + h^2) \approx 2xh + h^2 \)

In essence, this approximation simplifies and enables us to express \( \sin((x+h)^2) \) in a form that is easily inserted into the difference quotient. It drastically reduces the complexity when performing the limit: instead of grappling with multiple trigonometric terms, you can now directly observe terms that linearly contribute to the derivative expression. This aids in honing in on the precise influence of each small \( h \) change, ultimately making the derivation calculation smoothly approachable.