Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-18, find \(D_{x} y\). $$ y=2 \sin x+3 \cos x $$

Step-by-Step Solution

Answer

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

Step 1: Recall the Basic Derivatives

We need to find the derivative of the function \(y = 2 \sin x + 3 \cos x\). Recall that the derivative of \(\sin x\) with respect to \(x\) is \(\cos x\) and the derivative of \(\cos x\) is \(-\sin x\).

Step 2: Differentiate Each Term Separately

Apply the derivative to each term separately:- For the term \(2 \sin x\), the derivative is \(2 \cdot \cos x\).- For the term \(3 \cos x\), the derivative is \(3 \cdot (-\sin x) = -3 \sin x\).

Step 3: Combine the Derivatives

Combine the results from Step 2 to find the total derivative of the function:\(D_x y = 2 \cos x - 3 \sin x\).

Key Concepts

Basic Derivatives

Understanding basic derivatives is essential in calculus, especially when working with trigonometric functions like sine and cosine. The derivative reveals how a function changes as its input changes. For trigonometric functions:

• The derivative of \(\sin x\) with respect to \(x\) is \(\cos x\).
• The derivative of \(\cos x\) with respect to \(x\) is \(-\sin x\).

These basic derivatives form the cornerstone for more complex differentiation tasks. It's crucial to memorize these derivatives as they're frequently used in various calculus problems. This foundational knowledge allows us to tackle more advanced topics in calculus efficiently.
You should always start by identifying which basic functions you are working with and know their derivatives by heart. Once you have these, differentiating more complex functions becomes straightforward.

Differentiation Techniques

Differentiation is a key operation in calculus, allowing us to find the rate of change of a function. When you encounter a function composed of multiple terms, like our example function \(y = 2 \sin x + 3 \cos x\), a common technique is to differentiate each term separately. This simplifies the process significantly.Here's how it works:

• Identify each term independently within the equation.
• Apply the known basic derivative rule to each term.
• Keep the coefficients outside the differentiation untouched, only applying the derivative to the function itself.

For our example:- The term \(2 \sin x\) becomes \(2 \cdot \cos x\) when differentiated.- The term \(3 \cos x\) becomes \(-3 \cdot \sin x\) when differentiated.
The technique of splitting terms makes the differentiation process easier to handle and ensures accuracy. Mastering such techniques is fundamental for tackling complex calculus problems efficiently.

Calculus Problem Solving

Solving calculus problems often requires a strategic approach. You begin by clearly understanding the problem and then methodically applying the appropriate calculus principles to find a solution. Following a systematic approach can help in efficiently arriving at the correct answer.
In our exercise to find \(D_{x} y\), the approach was:

• Recall the basic derivatives needed for the function's components.
• Differentiating each term separately, utilizing their respective derivatives.
• Combining the differentiated terms to get the final derivative expression.

These steps form a clear pathway to solve calculus problems involving derivatives.
This reinforcement of both recall and application of derivatives showcases the necessity of both understanding the rules and employing them strategically. With practice, such structured problem-solving becomes intuitive, allowing you to tackle more complex problems with confidence.