Problem 1 - Calculus

Question

In Problems \(1-18\), 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: Understand the Problem

We are given a function \(y = 2 \sin x + 3 \cos x\) and need to find its derivative with respect to \(x\), represented as \(D_x y\).

Step 2: Apply the Power Rule and Constant Multipliers

The derivative of a constant times a function is the constant times the derivative of the function. We’ll separate the problem into two parts: \(D_x(2 \sin x)\) and \(D_x(3 \cos x)\).

Step 3: Differentiate \( \sin x\)

The derivative of \(\sin x\) with respect to \(x\) is \( \cos x\). Therefore, \(D_x (2 \sin x) = 2 \cdot \cos x\).

Step 4: Differentiate \( \cos x\)

The derivative of \(\cos x\) with respect to \(x\) is \(- \sin x\). Therefore, \(D_x (3 \cos x) = 3 \cdot (-\sin x) = -3 \sin x\).

Step 5: Combine Results

Combine the derivatives from Step 3 and Step 4 to find the overall derivative. Thus, \(D_x y = 2 \cos x - 3 \sin x\).

Key Concepts

Trigonometric Functions

Trigonometric functions play a significant role in calculus, especially when dealing with derivatives. These functions include sine (\( \sin x \)), cosine (\( \cos x \)), and tangent (\( \tan x \)), among others. In this exercise, we focus on the sine and cosine functions.

• The sine function, \( \sin x \), is a periodic function, meaning it repeats its pattern over regular intervals (every 360° or 2π radians).
• The cosine function, \( \cos x \), also exhibits periodic behavior with the same interval as the sine function.

These trigonometric functions are critical not only in geometry and trigonometry but also in calculus, providing solutions to real-world problems involving waves, oscillations, and many periodic phenomena.

Derivative Rules

Derivative rules are essential tools for differentiating functions. These rules simplify the process, making it more efficient and accurate. When dealing with a function like \(y = 2 \sin x + 3 \cos x\), you will apply different derivative rules to each component:

• The **Power Rule** states that the derivative of \(x^n\) is \(n \cdot x^{n-1}\). Though this rule does not directly apply here, its concept is fundamental.
• The **Constant Multiple Rule** is important here, which tells us that a constant multiplied by a function remains constant when the function is differentiated. For example, the derivative of \(2 \sin x\) is \(2 \cdot \frac{d}{dx}(\sin x)\).
• The **Sum Rule** allows us to differentiate each term separately. This means for \(D_x(2 \sin x + 3 \cos x)\), we can separately find the derivative of \(2 \sin x\) and \(3 \cos x\), then add them up.

By understanding and applying these rules, you can tackle more complex differentiation problems with ease.

Differentiation Techniques

Differentiation techniques are methods and strategies used to find the derivative of a function. In this exercise, systematic techniques help us break down the problem.

• First, isolate the components of the function: focus on one term at a time. Here, the function is split into \(D_x(2 \sin x)\) and \(D_x(3 \cos x)\). This organizes the differentiation process efficiently.
• Second, apply the relevant differentiation rules to each term. For \(D_x(2 \sin x)\), use the Constant Multiple Rule and the fact that the derivative of \(\sin x\) is \(\cos x\): you get \(2 \cdot \cos x\).
• For the term \(D_x(3 \cos x)\), apply the Constant Multiple Rule again along with the knowledge that the derivative of \(\cos x\) is \(-\sin x\): resulting in \(-3 \sin x\).

Lastly, combine the results from each term to give the complete derivative \(D_x y = 2 \cos x - 3 \sin x\). Remember, clear and structured techniques in differentiation can simplify even complex derivatives.