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The position function gives a mathematical description of an object's location at any given time. In our example, the position function is given by \( s(t) = \frac{60t}{\sqrt{t^2+1}} \). Here, \( s(t) \) represents the position at time \( t \). The numerator, \(60t\), indicates that position changes with time while the denominator, \(\sqrt{t^2+1}\), represents a factor that affects how the position changes.
This function essentially describes a path or trajectory. More deeply, it provides a framework for understanding how different aspects of motion, such as velocity and acceleration, relate to time.

• The numerator suggests linear movement, influenced by time \( t \).
• The denominator hints at a complex relationship, modifying the linear movement to account for the additional factor \( \sqrt{t^2+1} \).

Understanding the position function is crucial as it is the starting point for deriving other motion-related functions.

A derivative measures how a function changes when the input changes. In simpler terms, it gives the rate of change or the slope of the function at a particular point. When applied to a position function, the derivative yields the velocity function.
To move from the position function \( s(t) = \frac{60t}{\sqrt{t^2+1}} \) to its velocity counterpart, we differentiate with respect to time \( t \). This is done using calculus rules like the quotient and chain rules.

• The derivative \( s'(t) \) of our position function represents an instantaneous velocity at any given time \( t \).
• The process involves calculating limits, but here we skip to where the derivative simplifies to \( v(t) = \frac{60}{(t^2+1)^{3/2}} \).

Derivatives play a vital role in dynamics, enabling us to forecast how quickly an object moves and the path it might take over time.

The quotient rule is a technique in calculus used for differentiating functions that are represented as one function divided by another. Essentially, it is used when dealing with a ratio of two functions.
In our problem, the position function \( s(t) = \frac{60t}{\sqrt{t^2+1}} \) is a ratio of \( 60t \) and \( \sqrt{t^2+1} \).
The quotient rule is expressed as:\[\left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2}\]
where \( u = 60t \) and \( v = \sqrt{t^2+1} \). We apply this rule by:

• First finding the derivatives \( u' \) and \( v' \).
• Substituting those derivatives back into the formula.

This helps derive the rate of change which honors the structure of the original function and helps understand both the top (numerator) and bottom (denominator) contributions to the overall rate.

The chain rule is another powerful tool used in calculus to differentiate composite functions. That means when you have a function within a function, the chain rule helps determine how to take derivatives.
In our function \( s(t) = \frac{60t}{\sqrt{t^2 + 1}} \), the denominator involves a composition: \( \sqrt{t^2 + 1} \).
The chain rule is expressed as:\[(f(g(x)))' = f'(g(x)) \cdot g'(x)\]
This involves differentiating the outer function first while leaving the inner function unchanged, then multiplying by the derivative of the inner function.

• The outer function is \((t^2 + 1)^{-1/2}\), which differentiates accordingly.
• The inner function \( t^2 + 1 \) is simpler to differentiate resulting in \(2t\).

Applying the chain rule ensures all layers of complexity within a composite function are addressed, crucial for accurate differentiation of intricate functions.