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The derivative is a fundamental concept in calculus that describes how a function changes at a particular point. It is essentially the rate at which a function’s value changes as the input changes. Think of it as the function's rate of growth or decline. In our exercise, to find the derivative of the function \( f(t) = 6t^2 \), we are determining how rapidly \( f(t) \) changes with respect to changes in \( t \). Understanding derivatives allows you to analyze the behavior of functions over an interval and to find slopes of tangent lines at any point on a curve.
Derivatives are important in many fields such as physics, engineering, and economics because they can tell you about the velocity or acceleration of an object, the slope of financial trends, or even the growth rate of populations. This makes derivatives very versatile and an essential part of mathematics.

The power rule is a simple and powerful way to find derivatives, particularly when dealing with polynomial functions. This rule states that for any power function \( t^n \), the derivative is \( nt^{n-1} \). In simpler terms, you multiply the exponent by the coefficient and decrease the exponent by one. For example, if you have the function \( 6t^2 \), applying the power rule, you multiply 2 (the exponent) by 6 to get 12, and reduce the exponent by 1, resulting in \( 12t^{1} \). Hence, the derivative of \( 6t^2 \) is \( 12t \).
The power rule can make derivatives easy to compute, especially when dealing with simple monomials. By mastering this rule, you can quickly tackle problems involving polynomials, saving time and avoiding unnecessary errors.

• Identify the exponent of the variable.
• Multiply by the existing coefficient.
• Subtract one from the exponent.

This rule makes the task of finding derivatives straightforward and manageable, especially for students new to calculus.

Function simplification is an essential step in solving calculus problems. After computing derivatives or performing other operations on functions, you'll often end up with complex expressions that need to be simplified. Simplifying functions involves reducing fractions, combining like terms, or removing unnecessary components, making the expression easier to work with and understand.
In the provided exercise, after finding the derivative and computing the relative rate of change, we obtained the expression \( \frac{12t}{6t^2} \). By simplifying this expression, we divide both the numerator and the denominator by the greatest common factor, which is 6 here. This simplifies \( \frac{12t}{6t^2} \) to \( \frac{2}{t} \). Such simplifications transform cumbersome mathematical expressions into streamlined forms that are easier to interpret and apply to practical problems, such as finding specific rates of change.