Question

Use the quotient rule to find the derivative of the given function.

f(t)=7 t²-8 t/4 t+5

fʹ(t)=___

Answer 1

The derivative of the given function, using the quotient rule, is:

f'(t) = (72t - 8) / (4t + 5)

Step-by-step explanation:

To find the derivative of a function that involves division, we use the quotient rule. This rule states that if f(x) = g(x) / h(x), then f'(x) = [h(x)g'(x) - g(x)h'(x)] / [h(x)]².

In our case, let's define our functions as follows:

g(t) = 7t²

h(t) = 4t + 5

Now, we can calculate the derivatives of both functions using the power rule:

g'(t) = 14t

h'(t) = 4

Substituting these derivatives into the quotient rule formula, we get:

f'(t) = [(4t + 5)(14t) - (7t²)(4)] / [(4t + 5)]²

Simplifying this expression, we get our final answer:

f'(t) = (72t - 8) / (4t + 5)