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.