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Differentiation is a fundamental concept in calculus that involves finding the rate at which a function is changing at any given point. It allows us to determine how a function behaves and is particularly useful in solving problems related to motion, growth, and optimization. To differentiate a function, we use specific rules which depend on the type of function involved, such as polynomial, exponential, logarithmic, or trigonometric functions.
Differentiation is an operation that, when applied to a function, calculates the derivative. The derivative itself is a new function that gives the instantaneous rate of change of the original function. It helps in understanding how

• the function increases or decreases
• finding slopes of tangent lines
• identifying local maxima and minima

In the context of the original exercise, we are asked to find the first and second derivatives of the function involving trigonometric terms using differentiation rules specific to those forms.

The second derivative is essentially the derivative of the derivative. It provides deeper insight into the behavior of functions. While the first derivative gives us information about the slope or rate of change of the original function, the second derivative tells us how that rate of change itself is changing.
In practical terms:

• The second derivative can indicate concavity, helping us understand whether the graph of the original function is curving upwards or downwards.
• This concept is useful in applications such as determining points of inflection.
• It can identify acceleration in kinematics where the initial function represents position.

For the exercise, after obtaining the first derivative of the trigonometric function, we proceed to find its second derivative, which further informs us about the curvature of the graph represented by the function.

Trigonometric functions, often involving sine, cosine, and tangent, are essential tools in calculus. They appear frequently in problems involving waves, oscillations, and circular motion. The derivatives of these functions are has their specific patterns and rules, making them straightforward once you understand the fundamentals.
Some key principles include:

• The derivative of \( \sin(at) \) is \( a\cos(at) \).
• The derivative of \( \cos(bt) \) is \( -b\sin(bt) \).

In the context of our exercise, recognizing these derivatives is crucial for calculating the first and second derivatives efficiently. Applying these rules correctly helps to analyze and predict behavior based on trigonometric functions effectively.