Let y = 4t/(t^2 + 5). Find dy/dt, writing your answer in it's simplest form, and find all values of t for which dy/dt = 0
We use the quotient rule to find dy/dt. Let u = 4t and v = (t^2 + 5). Then, u' = 4 and v' = 2t. Hence,
dy/dt = u'v - v'u / v2 = 4(t^2 + 5) - 4t x 2t / (t^2 + 5)2 = 20 - 4t2 / (t^2 + 5)2. Now, we need to find all t such that dy/dt < 0 i.e.
20 - 4t2 / (t^2 + 5)2 < 0 which rearranges to give t2 > 5, so, t > 51/2 and t < -51/2.
JH
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