Differentiate the function.
y = ((4x^2 + 8x + 4)÷√(x))
x
the answer is (6√(x))+(4/√(x))-(2/x√(x))
Well, well, well, it looks like we have a differentiating challenge on our hands. Don't worry, I'm here to make it more fun! Let's put on our differentiation hats and get started.
First things first, we need to use the quotient rule to differentiate this function. And for that, we'll need the derivatives of both the numerator and denominator.
Let's start with the numerator, shall we? The derivative of 4x^2 + 8x + 4 is 8x + 8. Easy peasy lemon squeezy!
Now let's move on to the denominator, √(x). The derivative of √(x) is slightly trickier, but nothing we can't handle. Using the power rule, we get (1/2) * x^(-1/2). Woohoo!
Now, let's put it all together using the quotient rule. According to the quotient rule, the derivative of y = (4x^2 + 8x + 4) ÷ √(x) is:
[(8x + 8) * √(x) - (4x^2 + 8x + 4) * (1/2) * x^(-1/2)] / (√(x))^2
Simplifying this a bit further, we get:
[8x√(x) + 8√(x) - (2x^2 + 4x + 2) / 2√(x)]
So, the differentiated function is:
(8x√(x) + 8√(x) - x^2 - 2x - 1) / 2√(x)
I hope that brought a smile to your face, even though differentiation can be a bit of a clown fiesta sometimes!
To differentiate the function, we can use the quotient rule, which states that if we have a function in the form f(x) = g(x) / h(x), then the derivative of f(x) is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
In our case, g(x) = 4x^2 + 8x + 4 and h(x) = √(x), so we need to find g'(x) and h'(x) in order to apply the quotient rule.
First, let's find g'(x). Taking the derivative of g(x) term by term, we have:
g'(x) = d/dx (4x^2 + 8x + 4)
= 8x + 8
Next, let's find h'(x). Taking the derivative of h(x), we apply the chain rule since h(x) = √(x):
h'(x) = d/dx (√(x))
= (1/2) * (x)^(-1/2)
= 1 / (2√(x))
Now that we have g'(x) and h'(x), we can apply the quotient rule:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
= ((8x + 8) * √(x) - (4x^2 + 8x + 4) * (1 / (2√(x)))) / (√(x))^2
= (8x√(x) + 8√(x) - (4x^2 + 8x + 4) / (2√(x))) / x
Simplifying further, we have:
f'(x) = (8x√(x) + 8√(x) - (4x^2 + 8x + 4) / (2√(x))) / x
= (8x√(x) + 8√(x) - 4x^2 - 8x - 4) / (2x√(x))
Therefore, the derivative of the function y = ((4x^2 + 8x + 4) / √(x)) with respect to x is:
y' = (8x√(x) + 8√(x) - 4x^2 - 8x - 4) / (2x√(x))
Note: The function y = ((4x^2 + 8x + 4) / √(x)) has the restriction x > 0, since division by zero is undefined.
To differentiate the function y = ((4x^2 + 8x + 4)÷√(x))/x, we can use the quotient rule of differentiation. The quotient rule states that if we have a function in the form f(x)/g(x), then the derivative is given by:
y' = (g(x)f'(x) - f(x)g'(x))/[g(x)]^2
Let's break it down step by step:
1. Find f(x) and g(x):
f(x) = (4x^2 + 8x + 4)
g(x) = √(x)
2. Find f'(x) and g'(x):
Differentiating f(x) and g(x) gives us:
f'(x) = 8x + 8
g'(x) = 1/(2√(x))
3. Plug the values into the quotient rule formula:
y' = [√(x)*(8x + 8) - (4x^2 + 8x + 4)*(1/(2√(x))))]/[√(x)]^2
4. Simplify the expression:
To simplify, we can multiply the numerator and denominator by 2√(x) to remove the square root from the denominator:
y' = [2x(8x + 8) - (4x^2 + 8x + 4)]/[2x]
Expanding and simplifying the numerator:
y' = [16x^2 + 16x - 4x^2 - 8x - 4]/[2x]
y' = [12x^2 + 8x - 4]/[2x]
y' = (6x^2 + 4x - 2)/x
Thus, the derivative of the function y = ((4x^2 + 8x + 4)÷√(x))/x is y' = (6x^2 + 4x - 2)/x.