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The product rule is a fundamental concept in differentiation used when you have the product of two functions. If you have a function defined as the product of two separate functions, such as \( q(t) = f(t) \, g(t) \), then the derivative of \( q \) with respect to \( t \) is not just the product of the derivatives of \( f \) and \( g \). Instead, we use the product rule formula:

• \((fg)' = f' \, g + f \, g'\)

This means you differentiate \( f \), multiplying by \( g \), and then add \( f \) multiplied by the derivative of \( g \).
For instance, in the function \( q(t)=2e^{-t/2}\cos{2t} \), the term \( 2e^{-t/2} \) is \( f(t) \) and \( \cos{2t} \) is \( g(t) \). Thus, you differentiate each separately and combine using the product rule.
This ensures accurate computation of derivatives and helps in finding the rate of change for complex functions.

The chain rule is another essential differentiation technique used when dealing with compositions of functions. When you have a function nested within another, you apply the chain rule to differentiate effectively. The basic rule states:

• If \( h(t) = f(g(t)) \), then \( h'(t) = f'(g(t)) \, g'(t) \).

In this process, you first differentiate the outer function with respect to its inner function and then multiply by the derivative of the inner function.
For example, to differentiate \( 2e^{-t/2} \), identify that it's a composition of \( e^x \) and \( -t/2 \). First, differentiate \( e^{-t/2} \) as if -t/2 is the variable, then multiply by the derivative of \(-t/2\) with respect to \( t \).
This approach is particularly useful in trigonometric and exponential expressions within our function, ensuring all composite parts are handled correctly.

The rate of change of a function depicts how the function's value changes with respect to changes in its input variable. In calculus, this is represented as the derivative of the function. For the function \( q(t) = 2e^{-t/2} \cos{2t} \), the rate of change at a specific point, \( t = 1 \), is found by evaluating its derivative at that point.To calculate, you differentiate \( q(t) \) using the product and chain rules to obtain \( q'(t) \). Evaluating \( q'(t) \) at \( t = 1 \) gives the exact rate at which \( q(t) \)'s value is changing at this moment.
This concept is crucial in understanding physical phenomena like velocity or growth rates, where determining how a quantity evolves over time or another variable is needed.

Trigonometric functions are sine, cosine, and tangent, among others, which relate to angles and ratios within right-angled triangles. In calculus, their derivatives reveal how these functions change and oscillate.

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

When differentiating functions like \( \cos{2t} \), the change involved is found by first applying the derivative of \( \cos(x) \) and then considering the internal angle function with the chain rule.
As seen in \( g'(t) = -2\sin{2t} \), the derivative of \( \cos{2t} \) utilizes both its derivative formula and the consideration of the angle \( 2t \).
Understanding these principles is key in problems involving periodic motion or waves, where trigonometric functions often play a central role.