find the derivative using quotient rule

F(x) = (x - 7 x sqrt(x)) / sqrt(x)

F'(x) = [bottom*derivative of top - top *derivative of bottom] / bottom^2

=[x^.5 { 1-7(.5x x^-.5 +x^.5) } -(x-7x x^.5).5 x^-1.5 ]/ x
etc

To find the derivative of the function F(x) = (x - 7x√(x)) / √(x) using the quotient rule, you would need to follow these steps:

Step 1: Identify the numerator and denominator of the function F(x). In this case, the numerator is (x - 7x√(x)) and the denominator is √(x).

Step 2: Apply the quotient rule formula, which states that if you have a function h(x) = g(x) / f(x), the derivative h'(x) can be found using the formula:

h'(x) = [f(x) * g'(x) - g(x) * f'(x)] / [f(x)]^2

Step 3: Differentiate the numerator and denominator separately.

Differentiating the numerator:
For the term "x," the derivative is simply 1.
For the term "7x√(x)," you can use the product rule:
Let u = 7x and v = √(x).
The derivative of u is du/dx = 7 and the derivative of v is dv/dx = (1/2√(x)).
Using the product rule formula, the derivative of 7x√(x) is:
(du/dx * v) + (u * dv/dx) = (7 * √(x) + 7x * (1/2√(x)))
= 7√(x) + (7x/2√(x))
= 7√(x) + (7x^(3/2))/(2√(x))

Differentiating the denominator:
The derivative of √(x) is (1/2√(x)).
(Note: When differentiating √(x), you can think of it as x^(1/2), and apply the power rule.)

Step 4: Plug the derivatives of the numerator and denominator into the quotient rule formula.

h'(x) = [(√(x) * (7√(x) + (7x^(3/2))/(2√(x)))) - ((x - 7x√(x)) * (1/2√(x)))] / [√(x)]^2

Simplifying the formula:
h'(x) = [7x + (7x^(3/2))/(2√(x)) - (x/2)] / x

Step 5: Simplify the expression above as much as possible to obtain the final derivative of F(x).

h'(x) = [(14x - x + 7x^(3/2))/(2√(x))] / x
= (13x + 7x^(3/2)) / (2x√(x))

Therefore, the derivative of F(x) = (x - 7x√(x)) / √(x) using the quotient rule is:
F'(x) = (13x + 7x^(3/2)) / (2x√(x))