To find the derivative of the function f(t) = gt(3t) using the definition of the derivative, we start by recalling what the definition entails. The derivative of a function at a point is defined as:
f'(t) = lim (h -> 0) [(f(t+h) – f(t)) / h]
We can break this down into steps. Let’s first compute f(t + h).
Substituting into our function, we get:
f(t + h) = gt(3(t + h)) = gt(3t + 3h)
Now we subtract f(t) from f(t + h):
f(t + h) – f(t) = gt(3t + 3h) – gt(3t)
Next, place this difference into our derivative formula:
f'(t) = lim (h -> 0) [(gt(3t + 3h) – gt(3t)) / h]
To solve this limit, we need to analyze the behavior of the expression as h approaches 0. Depending on the nature of the function gt, we could use properties of limits and possibly derivative rules if gt is standard.
Assuming gt is a differentiable function, we could apply the chain rule here, particularly if it follows the basic derivative rules. We can anticipate that:
f'(t) = g'(t) * d/dt(3t) = 3g'(t)
Thus, the derivative of the function gt(3t) at the point t is:
f'(t) = 3g'(3t)
This shows how we can use the definition of the derivative step by step while incorporating the rules of differentiation. It emphasizes the importance of understanding function behavior in calculus.