Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=2 \sin x+3 \cos x $$

Step-by-Step Solution

Answer

The derivative is \( D_x y = 2\cos x - 3\sin x \).

Step 1: Identify the components of the function

The given function is \( y = 2\sin x + 3\cos x \). This function is a combination of two trigonometric functions: \( 2\sin x \) and \( 3\cos x \).

Step 2: Differentiate each component separately

Find the derivative of \( 2\sin x \) and \( 3\cos x \). Using the derivative rules for sine and cosine:\[\frac{d}{dx}(\sin x) = \cos x \quad \text{and} \quad \frac{d}{dx}(\cos x) = -\sin x\]Calculate the derivatives:\[\frac{d}{dx}(2\sin x) = 2\cos x\quad \text{and} \quad \frac{d}{dx}(3\cos x) = -3\sin x\]

Step 3: Add the results of the derivatives

Combine the derivatives obtained from Step 2. The derivative of \( y \), denoted as \( D_x y \), is the sum of these individual derivatives:\[D_x y = 2\cos x - 3\sin x\]

Step 4: Finalize the derivative

The final derivative of the function \( y = 2\sin x + 3\cos x \) is:\[ D_x y = 2\cos x - 3\sin x \]

Key Concepts

Differentiation Techniques

Differentiation is a core concept in calculus, allowing us to discover the rate of change of a function. It's essential in various fields, from physics to economics. When differentiating trigonometric functions, understanding the basic rules helps make the process smoother.
For our function, which is a combination of sine and cosine terms, the key differentiation technique involves applying known derivatives:

• The derivative of the sine function is the cosine function, which means that the rate at which \(\sin x\) changes over x is given by \(\cos x\).
• Conversely, the derivative of the cosine function is negative sine. Thus, the rate at which \(\cos x\) changes is given by \(-\sin x\).

Using these basic rules allows us to handle more complex expressions like \(2\sin x + 3\cos x\) by differentiating each term separately and then combining the results.

Trigonometric Identities

Understanding trigonometric identities is crucial when working with trigonometric functions and their derivatives. These identities help simplify calculations and solve problems that involve angles.
For differentiation, some common trigonometric identities include:

• The Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\), which relates the sine and cosine of the same angle.
• The angle sum and difference identities, which allow us to express trigonometric functions of combined angles (e.g., \(\sin(a + b)\) and \(\cos(a - b)\)) in terms of products of sine and cosine.
• The fundamental derivatives: \(\frac{d}{dx}(\sin x) = \cos x\) and \(\frac{d}{dx}(\cos x) = -\sin x\), which provide a foundation for finding derivatives of more complicated trigonometric expressions.

By leveraging these identities and derivatives, students can easily tackle problems involving trigonometric functions and derivatives.

Calculus Problem-Solving

Solving calculus problems, especially those involving derivatives, requires a structured approach. It's crucial to break down the problem into smaller, manageable parts while being aware of the fundamental rules and identities.
In our example with \(y = 2\sin x + 3\cos x\), understanding multiplication and addition rules in derivatives was key:

• Differentiate each part separately. If a function is composed of multiple parts, tackle each one individually. For \(2\sin x\), the derivative is \(2\cos x\), and for \(3\cos x\), the derivative is \(-3\sin x\).
• Combine results: Once you have the individual derivatives, sum them up to get the overall derivative: \(D_x y = 2\cos x - 3\sin x\).
• Check your results. Always verify using basic rules to ensure accuracy.

Approaching problems step-by-step not only makes the process less daunting but also allows for a deeper understanding of the underlying calculus principles.