How do you find the derivative of the function g(t) = 9t using the definition of a derivative?

To find the derivative of the function g(t) = 9t using the definition of derivative, we start with the formal definition:

Definition of Derivative: The derivative of a function g(t) at a point t is defined as:

g'(t) = lim (h -> 0) [(g(t + h) – g(t)) / h]

1. Step 1: Substitute the function into the definition.
   For our function, we need to compute g(t + h).
2. First, substitute t + h into the function:

g(t + h) = 9(t + h) = 9t + 9h

3. Step 2: Compute g(t + h) – g(t).
   Now, we can find:

g(t + h) – g(t) = (9t + 9h) – (9t) = 9h

4. Step 3: Substitute into the limit.
   Now, plug this back into our limit definition:

g'(t) = lim (h -> 0) [(9h) / h]

5. Notice that h in the numerator and denominator cancels out (as long as h ≠ 0):

g'(t) = lim (h -> 0) [9]

6. Step 4: Evaluate the limit.
   Since the expression does not depend on h anymore, we can evaluate the limit:

g'(t) = 9

Thus, the derivative of the function g(t) = 9t is g'(t) = 9.