#### To determine

**To find:** The derivative of the function f(t)=2.5t2+6t and state the domain of the function and its derivative.

#### Answer

The derivative of the function f(t) is 5t+6.

The domain of the function f(t) is ℝ

The domain of f′(t) is ℝ.

#### Explanation

**Formula used:**

The derivative of a function *f ,* denoted by f′(x), is

f′(x)=limh→0f(x+h)−f(x)h (1)

**Calculation:**

Obtain the derivative of the function f(t).

Compute f′(t) by using the equation (1),

f′(t)=limh→0f(t+h)−f(t)h=limh→0(2.5(t+h)2+6(t+h))−(2.5t2+6t)h=limh→0(2.5(t2+h2+2xh)+6t+6h)−2.5t2−6th=limh→02.5t2+2.5h2+5th+6t+6h−2.5t2−6th

Simplify the numerator,

f′(t)=limh→02.5h2+5th+6hh=limh→0h(2.5h+5t+6)h

Since the limit *h* approaches zero but not equal to zero, cancel the common term *h* from both the numerator and the denominator,

f′(t)=limh→0(2.5h+5t+6)=2.5(0)+5t+6=5t+6

Thus, the derivative of the function f(t) is 5t+6.

The domain of the function f(t) is ℝ since every t∈ℝ there is unique image f(t)∈ℝ.

The domain of the derivative f′(t) is {t∈ℝ| f′(t) exists}.

Since the derivative f′(t)=5t+6 exists for all real numbers.

Therefore, the domain of f′(t) is ℝ.