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The product rule is a key concept in calculus for finding the derivative of a product of two functions. Essentially, it allows you to differentiate a product of two functions without having to multiply them out first, which can simplify your work considerably. The product rule states that if you have two functions, say \( u(x) \) and \( v(x) \), then the derivative of their product is:

• \((uv)' = u'v + uv'\)

This formula makes it easier to handle products. You differentiate one function while keeping the other function unchanged, then change roles. Combine the two results to find the desired derivative.
In the exercise, we applied the product rule to the functions \( e^{3x} \) and \( \cos^2(7x) \). Remember, the product rule is an invaluable tool for tackling a variety of derivative problems in calculus that involve multiplying functions together.

The chain rule is a calculus essential, especially when dealing with composite functions—functions within other functions. It's what you use to differentiate a function of a function, replacing the need to unravel complex chains of operations manually. The chain rule can be expressed as:

• \((f(g(x)))' = f'(g(x)) \cdot g'(x)\)

In simpler terms, you differentiate the outer function first, then multiply by the derivative of the inner function.
In our specific exercise, the chain rule was used twice. We applied it to handle the derivatives of \( e^{3x} \) and \( \cos^2(7x) \). The derivative of \( e^{3x} \) was calculated as \( 3e^{3x} \), thanks to the chain rule. Similarly, \( \cos^2(7x) \) required using the chain rule to reach the derivative \( -14\cos(7x)\sin(7x) \).
Remember, whenever you see a composite function (one function inside another), the chain rule is what you need to properly find its derivative.

Exponential functions are a core part of calculus and a variety of applications in science and engineering. An exponential function is of the form \( a^{x} \), where \( a \) is a constant, and \( x \) is the variable. These functions are unique because they grow rapidly, and their derivatives are proportional to themselves.
For instance, when you differentiate \( e^{x} \), the result is simply \( e^{x} \). When combined with other expressions, such as \( e^{3x} \), the derivative is found using the chain rule, resulting in \( 3e^{3x} \).
Exponential growth and decay model many natural phenomena, like population growth or radioactive decay, making understanding these functions crucial.

Trigonometric functions include sine, cosine, and tangent, fundamental tools in calculus. These functions are periodic, with values that repeat at regular intervals, often used in modeling waves and oscillations.
In our exercise, the function \( \cos^2(7x) \) was differentiated. The power of 2 combined with the angle \( 7x \) made it complex, requiring the chain rule for differentiation. The result, \( -14\cos(7x)\sin(7x) \), reflects the derivative of the cosine squared and the angle's impact.
Understanding how to work with trigonometric functions' derivatives expands your calculus toolbox, as these functions appear in various fields, including physics and engineering.