Find the derivative of the function below in two ways.

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

(a) By using the Quotient Rule.
(b) By simplifying first.

Can't you do either of these? Do you know the quotient rule?

If you simplify first, you get
F(x) = sqrt x - 7x

Try differentiating that term by term. There are only two terms.

Someone will gladly critique your work.

To find the derivative of the function F(x) in two ways, we will first use the Quotient Rule and then simplify the expression before taking the derivative.

(a) By using the Quotient Rule:
The Quotient Rule states that if we have a function F(x) = u(x) / v(x), where u(x) and v(x) are differentiable functions, then the derivative of F(x) with respect to x is given by:

F'(x) = (v(x) * u'(x) - u(x) * v'(x)) / [v(x)]^2

Applying the Quotient Rule to our function F(x) = (x - 7√(x)) / √(x), we can assign u(x) = x - 7√(x) and v(x) = √(x):

u'(x) = 1 - 7 * (1/2) * √(x) = 1 - (7/2) * √(x)

v'(x) = (1/2) * x^(-1/2) = 1 / (2√(x))

Now, substituting u(x), u'(x), v(x), and v'(x) into the Quotient Rule formula:

F'(x) = [(√(x) * (1 - (7/2) * √(x))) - ((x - 7√(x)) * (1 / (2√(x))))] / [√(x)]^2

Simplifying the expression in the brackets and the denominator:

F'(x) = [(√(x) - (7/2 * x)) - (x / (2√(x)) - (7/2)) / [√(x)]^2

Combining like terms and simplifying further:

F'(x) = [(2√(x) - 7x - x + 7) / 2√(x)] / [√(x)]^2

F'(x) = [(-6x + 7 + 2√(x) - 7) / 2√(x)] / [√(x)]^2

F'(x) = (-6x + 2√(x)) / [2√(x) * √(x)]

F'(x) = (-6x + 2√(x)) / (2x)

Simplifying further:

F'(x) = -3x + √(x) / x

(b) By simplifying first:
Before using the Quotient Rule, we can simplify the expression (x - 7√(x)) / √(x) by multiplying the numerator and denominator by √(x):

F(x) = [(x - 7√(x)) * √(x)] / [√(x) * √(x)]

F(x) = (x√(x) - 7x) / x

Simplifying the expression further:

F(x) = √(x) - 7

Now, we can easily find the derivative of F(x) = √(x) - 7:

F'(x) = d/dx (√(x) - 7)

F'(x) = (1/2) * x^(-1/2) - 0

F'(x) = (1/2√(x))

So, the derivative of the function F(x) using both methods is:
(a) F'(x) = -3x + √(x) / x
(b) F'(x) = 1/2√(x)