To determine

To find: The derivative of the function f(x)=x2−2x3 and state the domain of the function and its derivative.

Answer

The derivative of the function f(x) is 2x−6x2.

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 the equation (1),

f′(x)=limh→0f(x+h)−f(x)h=limh→0((x+h)2−2(x+h)3)−(x2−2x3)h=limh→0(x2+h2+2xh−2(x3+h3+3x2h+3xh2))−(x2−2x3)h=limh→0x2+h2+2xh−2x3−2h3−6x2h−6xh2−x2+2x3h

Simplify the numerator,

f′(x)=limh→0h2+2xh−2h3−6x2h−6xh2h=limh→0h(h+2x−2h2−6x2−6xh)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′(x)=limh→0(h+2x−2h2−6x2−6xh)=((0)+2x−2(0)2−6x2−6x(0))=2x−6x2

Thus, the derivative of the function f(x) is 2x−6x2.

The domain of the function f(x) is ℝ since 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)=2x−6x2 exists for all real numbers.

Therefore, the domain of f′(x) is ℝ.