Final answer:
The derivative of the function f(t) = 7.5t² - 5t is f'(t) = 15t - 5.
Step-by-step explanation:
To find the derivative of the function f(t) = 7.5t² - 5t using the definition of derivative, we can use the limit definition of derivative:
f'(t) = lim(h → 0) (f(t + h) - f(t)) / h
Let's substitute the given function:
f'(t) = lim(h → 0) (7.5(t + h)² - 5(t + h) - 7.5t² + 5t) / h
Simplifying the expression, we get:
f'(t) = lim(h → 0) (7.5t² + 15th + 7.5h² - 5t - 5h - 7.5t² + 5t) / h
Cancelling out the terms, we are left with:
f'(t) = lim(h → 0) (15th + 7.5h² - 5h) / h
Now, we can simplify further:
f'(t) = lim(h → 0) 15t + 7.5h - 5
Since we are taking the limit as h approaches 0, the term 7.5h and any other terms dependent on h will become 0. Hence, the derivative of the given function is:
f'(t) = 15t - 5