Question

F(x)=2x√x-1/(√x); f`(4)=?

Answer 1

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Answer 2

To find the derivative of the given function, we can use the quotient rule. The quotient rule states that for functions u(x) and v(x), where f(x) = u(x)/v(x), the derivative is given by:

f'(x) = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2

Let's apply the quotient rule to the given function F(x) = 2x√(x - 1)/√x.

First, let's find the derivative of the numerator u(x) = 2x√(x - 1):

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

Next, let's find the derivative of the denominator v(x) = √x:

v'(x) = (1/2) * x^(-1/2) = 1 / (2√x)

Now, we can substitute these values into the quotient rule formula:

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

Now, let's evaluate f'(4) by substituting x = 4 into the derivative expression:

f'(4) = (2 * 4 * (4 - 1) + 4) / (√(4 - 1) * √4) = (8 * 3 + 4) / (√3 * 2) = (24 + 4) / (√3 * 2) = 28 / (√3 * 2)

Therefore, f'(4) = 14/√3 or approximately 8.08 (rounded to two decimal places).