Differentiate the function.

y = (5x^2+3x+3)/sqrt.(x)

[ bottom*derivative of top- top*derivative of bottom ]/bottom^2

f(x)=1/✔x+5. En x=4

✔= raiz cuadrada

To differentiate the given function, y = (5x^2 + 3x + 3) / sqrt(x), we will use the quotient rule of differentiation. The quotient rule states that if we have a function in the form of f(x) / g(x), then its derivative is given by:

dy/dx = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2

Now let's apply the quotient rule to differentiate the given function step by step:

1. Identify f(x) and g(x):

f(x) = 5x^2 + 3x + 3
g(x) = sqrt(x)

2. Find f'(x) and g'(x):

f'(x) = d/dx(5x^2 + 3x + 3) = 10x + 3
g'(x) = d/dx(sqrt(x)) = 1 / (2 * sqrt(x))

3. Apply the quotient rule:

dy/dx = (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2

dy/dx = (sqrt(x) * (10x + 3) - (5x^2 + 3x + 3) * (1 / (2 * sqrt(x)))) / (sqrt(x))^2

4. Simplify the expression:

dy/dx = (10x * sqrt(x) + 3 * sqrt(x) - (5x^2 + 3x + 3) * (1 / (2 * sqrt(x)))) / x

dy/dx = (10x * sqrt(x) + 3 * sqrt(x) - (5x^2 + 3x + 3) / (2 * sqrt(x))) / x

dy/dx = (20x^2 + 6x - (5x^2 + 3x + 3)) / (2 * x * sqrt(x))

dy/dx = (20x^2 + 6x - 5x^2 - 3x - 3) / (2 * x * sqrt(x))

dy/dx = (15x^2 + 3x - 3) / (2 * x * sqrt(x))

Therefore, the derivative of the given function y = (5x^2 + 3x + 3) / sqrt(x) is dy/dx = (15x^2 + 3x - 3) / (2 * x * sqrt(x)).

To differentiate the function y = (5x^2 + 3x + 3) / √(x), we need to follow a few steps. Let's break it down:

Step 1: Check if the function is defined for the entire domain.
Since we have √(x) in the denominator, we need to ensure that x is greater than 0 to avoid division by zero. Therefore, the domain of this function is x > 0.

Step 2: Simplify the function if possible.
We notice that there is no way to simplify the given function further.

Step 3: Apply the Quotient Rule.
The Quotient Rule is used for differentiating functions in the form of (f(x) / g(x)). According to the Quotient Rule, if we have y = f(x) / g(x), the derivative of y with respect to x is given by:

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

In our case, f(x) = (5x^2 + 3x + 3) and g(x) = √(x). Therefore, f'(x) refers to the derivative of f(x) and g'(x) refers to the derivative of g(x).

Step 4: Differentiate f(x).
To differentiate f(x), we will use the power rule and the sum rule. The power rule states that for any real number n, d/dx(x^n) = n * x^(n-1). The sum rule states that the derivative of a sum of functions is equal to the sum of their derivatives. Applying these rules to f(x):

f'(x) = d/dx(5x^2) + d/dx(3x) + d/dx(3)
= 10x + 3 + 0
= 10x + 3

Step 5: Differentiate g(x).
To differentiate g(x) = √(x), we can rewrite it in terms of exponents. g(x) = x^(1/2). Applying the power rule:

g'(x) = d/dx(x^(1/2))
= (1/2) * x^(-1/2)
= (1/2) / (sqrt.(x))

Step 6: Substitute back into the Quotient Rule.
Now that we have f'(x) = 10x + 3 and g'(x) = (1/2) / (sqrt.(x)), we can substitute these back into the Quotient Rule:

dy/dx = [(√(x) * (10x + 3)) - ((5x^2 + 3x + 3) * (1/2))/(sqrt.(x))^2]
= (2(10x + 3) - (5x^2 + 3x + 3))/(2 * √(x))^2
= (20x + 6 - 5x^2 - 3x - 3)/(4x)
= (-5x^2 + 17x + 3)/(4x)

Therefore, the derivative of y = (5x^2 + 3x + 3) / √(x) with respect to x is dy/dx = (-5x^2 + 17x + 3) / (4x).