Problem 21 - Calculus of a Single Variable

Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=x^{2}-\frac{1}{2} \cos x $$

Step-by-Step Solution

Answer

The derivative of the function \( y = x^{2} - \frac{1}{2} \cos x \) is \( y' = 2x + \frac{1}{2} \sin x \).

Step 1: Differentiate the power function

The function \( y \) involves a term \( x^{2} \), which is a power function. The rule for the derivative of a power function \( x^{n} \) is \( nx^{n-1} \). Here, \( n \) is 2. Therefore, applying the rule to \( x^{2} \), we get the derivative as \( 2x^{2-1} \) or \( 2x \).

Step 2: Differentiate the cosine function

The function \( y \) also has a term \( -\frac{1}{2} \cos x \), which includes a cosine function. The derivative of \( \cos x \) is \( -\sin x \). However, since it is multiplied by \( -\frac{1}{2} \), the derivative becomes \(-(-\frac{1}{2} \sin x) \), which simplifies to \( \frac{1}{2} \sin x \).

Step 3: Combine the derivatives

Now that both parts of the function have been differentiated, they need to be combined to get the derivative of the overall function. Combining the derivatives from Steps 1 and 2, we get the derivative of \( y \) as \( 2x + \frac{1}{2} \sin x \).

Key Concepts

Power Rule

The power rule is a fundamental technique used in differentiation, particularly useful when dealing with polynomials. When differentiating a term like \(x^n\), the power rule tells us to bring the exponent down as a coefficient and then subtract one from the exponent. For instance, in our exercise, we have the term \(x^2\). Applying the power rule,

• Bring down the exponent: 2.
• Subtract one from the exponent: \(2 - 1 = 1\).

This results in the derivative \(2x\) for the term \(x^2\).
Utilizing the power rule simplifies the process of finding derivatives and is especially handy when working with multiple terms in an expression.

Trigonometric Functions

Trigonometric functions like sine and cosine have specific derivatives that are always true, regardless of the function's composition. These derivatives are essential in calculus.
For cosine, the derivative is

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

This pattern holds true for any occurrence of cosine in a function. By knowing these relationships, we can quickly find derivatives involving trigonometric functions, which are pivotal in many real-world applications, such as physics and engineering.

Derivative of Cosine

The derivative of the cosine function is an important rule in calculus. Specifically, the derivative of \(\cos x\) is \(-\sin x\). In the given exercise, the term \(-\frac{1}{2} \cos x\) needs to be differentiated. Since the derivative of \(\cos x\) is \(-\sin x\),

• Multiplying this by \(-\frac{1}{2}\) results in \(-(-\frac{1}{2} \sin x)\).

This simplifies to \(\frac{1}{2} \sin x\). Understanding how multiplication affects trigonometric derivatives is crucial to managing complex calculus problems.

Combining Derivatives

Once each part of the function is differentiated separately, the next step is to combine these derivatives. This process involves simply adding or subtracting the results from each part of the function.
In our exercise, after differentiating \(x^2\) to get \(2x\) and \(-\frac{1}{2} \cos x\) to get \(\frac{1}{2} \sin x\), we combine them:

• Add the derivative of the power function: \(2x\).
• Add the derivative of the trigonometric function: \(\frac{1}{2} \sin x\).

Thus, the combined derivative of the function \(y\) is \(2x + \frac{1}{2} \sin x\). Combining derivatives is straightforward yet essential when finding the complete derivative of multifaceted functions.