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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

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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

01

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 \( 2x \).
02

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 \( \frac{1}{2} \sin x \).
03

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

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Most popular questions from this chapter

Electricity The combined electrical resistance \(R\) of two resistors \(R_{1}\) and \(R_{2},\) connected in parallel, is given by $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$$ where \(R, R_{1},\) and \(R_{2}\) are measured in ohms. \(R_{1}\) and \(R_{2}\) are increasing at rates of 1 and 1.5 ohms per second, respectively. At what rate is \(R\) changing when \(R_{1}=50\) ohms and \(R_{2}=75\) ohms?

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