Question

Consider the function (f(x) = vž - 8). Find (f'(x)) and determine its domain.

a) (f'(x) = vž), Domain: All real numbers
b) (f'(x) = vž), Domain: (x in mathbb{R}) except (x = 0)
c) (f'(x) = 1), Domain: (x in mathbb{R})
d) (f'(x) = 0), Domain: No real solutions

Answer 1

Option B). To find the derivative of the function f(x) = √x - 8, we can use the power rule of differentiation. The power rule states that the derivative of x^n is equal to n*x^(n-1). The domain of the derivative f'(x) is all real numbers greater than or equal to 0, except x = 0.

Step-by-step explanation:

To find the derivative of the function f(x) = √x - 8, we can use the power rule of differentiation. The power rule states that the derivative of x^n is equal to n*x^(n-1). In this case, n is equal to 1/2 because the square root can be represented as x^(1/2). So, the derivative of √x is equal to 1/2*x^(-1/2).

To determine the domain of f'(x), we need to look at the domain of the original function f(x). Since the square root function is defined for all non-negative real numbers, the domain of f(x) is all real numbers greater than or equal to 0. Therefore, the domain of f'(x) is also all real numbers greater than or equal to 0. So, the correct answer is:

b) f'(x) = √x, Domain: (x ∈ ℝ) except (x = 0)