Question

Find the derivative of y = (x² - 8x + 3) / √x:

A. (2x - 8) / (2√x)
B. (2x - 8) / (x√x)
C. (2x - 8) / (3√x)
D. (2x - 8) / (4√x)

Answer 1

The derivative of y = (x² - 8x + 3) / √x is (2x - 8) / (2√x), which corresponds to option A.

Step-by-step explanation:

To find the derivative of y = (x² - 8x + 3) / √x, we need to apply the quotient rule which states that if we have a function y = f(x)/g(x), its derivative y' is given by (f'(x)g(x) - f(x)g'(x))/(g(x))^2. In this case, f(x) = x² - 8x + 3 and g(x) = √x.

The derivatives of these functions are f'(x) = 2x - 8 and g'(x) = 1/(2√x), since the derivative of x^(1/2) is (1/2)x^(-1/2). Therefore, we can plug these into our quotient rule formula and simplify as follows:

(f'(x)g(x) - f(x)g'(x))/(g(x))^2 = ((2x - 8)√x - (x² - 8x + 3)(1/(2√x)))/x

After simplifying, we find that y' = (2x - 8) / (2√x), which corresponds to option A.