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The product rule is a fundamental rule in calculus used to find the derivative of the product of two functions. When you have a function that is the multiplication of two smaller functions, we denote them as \( u \) and \( v \). To apply the product rule,

• Find the derivative of each function separately, called \( u' \) and \( v' \).
• Then, calculate the derivative of the whole expression using the formula: \( \frac{dy}{dx} = u'v + uv' \).

In our case, where \( y = x \cos x \), \( u = x \) and \( v = \cos x \). Therefore, the derivatives are \( u' = 1 \) and \( v' = -\sin x \). Using the product rule, \[ \frac{dy}{dx} = 1 \cdot \cos x + x \cdot (-\sin x) = \cos x - x \sin x. \]

Differentiation is the process of calculating the derivative of a function, which represents the rate at which the function's value changes. The first derivative provides insights into the slope of the function's graph. Here,

• The first derivative tells us how the product \( x \cos x \) changes as \( x \) changes.
• For more complex functions, such as a product, we use methods like the product rule.

Once the first derivative is found, differentiation can be repeated to find higher order derivatives, like the second derivative. The second derivative, \( \frac{d^2y}{dx^2} \), gives us insights into the concavity or the acceleration of change of the function. In our exercise, by differentiating \( \cos x - x \sin x \) again, we get the second derivative:\[ \frac{d^{2}y}{dx^{2}} = -2 \sin x - x \cos x. \] This expression indicates how the rate of change is itself changing.

Trigonometric functions, such as sine and cosine, are fundamental in calculus and have special rules for differentiation. The cosine function, \( \cos x \), and sine function, \( \sin x \), behave predictably:

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

These rules are vital when differentiating products involving trigonometric terms. Consider our function \( y = x \cos x \):

• \( v = \cos x \) contributes a \( -\sin x \) in the first derivative.
• Then, when finding the second derivative, both \( \sin x \) and \( \cos x \) appear, showing their influence in the overall rate of change of the function.

Understanding the behavior of these functions is essential for mastering how changes in one variable affect a given trigonometric expression.