#### To determine

**To find:** The derivative of the function f(x)=x4 and state the domain of the function and its derivative.

#### Answer

The derivative of the function f(x) is 4x3.

The domain of the function f(x) is ℝ

The domain of f′(x) 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(x).

Compute f′(x) by using equation (1) as follows.

f′(x)=limh→0f(x+h)−f(x)h=limh→0(x+h)4−x4h=limh→0(x4+4x3h+6x2h2+4xh3+h4)−x4h=limh→0x4+4x3h+6x2h2+4xh3+h4−x4h

Simplify the numerator,

f′(x)=limh→04x3h+6x2h2+4xh3+h4h=limh→0h(4x3+6x2h+4xh2+h3)h

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

f′(x)=limh→0(4x3+6x2h+4xh2+h3)=(4x3+6x2(0)+4x(0)2+(0)3)=4x3

Thus, the derivative of the function f(x) is 4x3.

The function f(x)=x4 is defined for every real (−∞,∞).

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

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

Since the derivative f′(x)=4x3 exists for all real numbers, the domain of f′(x) is ℝ.

Therefore, the derivative of the function f(x) is 4x3, domain of the function f(x) is ℝ and the domain of f′(x) is ℝ.