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The derivative of a function represents the rate at which the function value changes as its input changes. In more graphical terms, the derivative can be viewed as the slope of the tangent line to the function's graph at any given point.
For example, if you have a function that describes the distance a car travels over time, the derivative of that function at any point will give you the speed of the car at that specific time.

Finding a Derivative

When we found the derivative of the function s(t), we were essentially looking for the rate at which s(t) changes with respect to t. Moreover, whenever dealing with more complex functions like s(t) = (t/(2t + 1))^(3/2), which include compositions of functions or fractions, we need to apply specific rules like the chain rule and the quotient rule.

The chain rule is a formula for computing the derivative of the composition of two or more functions. The rule essentially states that if you have a function h(x) = g(f(x)), the derivative of h with respect to x can be found by taking the derivative of g with respect to f(x), and then multiplying it by the derivative of f with respect to x.
For our exercise, where s(t) = (t/(2t + 1))^(3/2), the function s(t) can be seen as a composition of the outer function u^(3/2) and the inner function u = t/(2t + 1). In Step 1 of the solution, we differentiated the outer function with respect to u, and in Step 2, we found the derivative of the inner function with respect to t, which set us up to apply the chain rule in Step 3, where we multiplied both derivatives to find the final derivative of s(t) with respect to t.

When we have a function represented as a ratio of two other functions, like u(t) = t/(2t + 1), we use the quotient rule to find the derivative.
The rule is given by the formula: for the function q(t) = f(t)/g(t), the derivative q'(t) is calculated as (f'(t)g(t) - f(t)g'(t))/(g(t))^2. This formula stems from the product rule and the chain rule.

Applying the Quotient Rule

In Step 2 of our exercise, the quotient rule was used to differentiate the inner function of our overall expression. The numerator function, f(t), was t, and the denominator function, g(t), was (2t + 1). We differentiated each separately and then applied the quotient rule to obtain the derivative of the inner function, u(t), with respect to t. The proper application of this rule is crucial when dealing with fractions within a function, as it allows us to accurately capture the rate of change of the ratio as a whole.