How can I find the derivative of the function f(t) = 7t using the definition of the derivative?

To find the derivative of the function f(t) = 7t using the formal definition of a derivative, we start with the definition itself:

The derivative of a function f(t) at a point t is defined as:

D = lim (h → 0) [f(t + h) - f(t)] / h

Now, let’s apply this definition step by step for the function f(t) = 7t:

1. Calculate f(t + h):

   Substituting t + h into the function:

   f(t + h) = 7(t + h) = 7t + 7h

2. Calculate f(t + h) – f(t):

   Subtract f(t) from f(t + h):

   f(t + h) - f(t) = (7t + 7h) - (7t) = 7h

3. Now we can substitute this back into the derivative formula:

D = lim (h → 0) [7h] / h

4. Simplify the expression:

   D = lim (h → 0) 7

5. Finally, take the limit as h approaches 0:

   D = 7

Therefore, the derivative of the function f(t) = 7t is:

f'(t) = 7

This result shows that the function f(t) = 7t has a constant slope of 7, indicating that it is a linear function with a consistent rate of change.