Preview Activity 2.3.1.
(a)
(b)
Let \(p(t) = 2t^2 (t^3 + 4t)\) and observe that \(p(t) = f(t) \cdot g(t)\text{.}\) Rewrite the formula for \(p\) by distributing the \(2t^2\) term. Then, compute \(p'(t)\) using the sum and constant multiple rules.
(c)
True or false: \(p'(t) = f'(t) \cdot g'(t)\text{.}\) Why?
(d)
Let \(q(t) = \frac{t^3 + 4t}{2t^2}\) and observe that \(q(t) = \frac{g(t)}{f(t)}\text{.}\) Rewrite the formula for \(q\) by dividing each term in the numerator by the denominator and simplify to write \(q\) as a sum of constant multiples of powers of \(t\text{.}\) Then, compute \(q'(t)\) using the sum and constant multiple rules.
(e)
True or false: \(q'(t) = \frac{g'(t)}{f'(t)}\text{.}\) Why?
