Question

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

Short answer

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

Step-by-step solution

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 \( 2x \).

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 \( \frac{1}{2} \sin x \).

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 \( 2x + \frac{1}{2} \sin x \).