hi i am having trouble solving this example of finding the derivate using the quotient rule?

y=(-4x^2+8)/(x^2+2)^2

what do u do with the (x^2+2)^2. please i get the wrong answer while factoring thanks.

To find the derivative of the given function using the quotient rule, you need to apply the following steps:

Step 1: Identify the numerator and denominator of the function.
In this case, the numerator is -4x^2 + 8, and the denominator is (x^2 + 2)^2.

Step 2: Apply the quotient rule formula.
The quotient rule states that if you have a function y = f(x)/g(x), then the derivative of y with respect to x is given by the formula:

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

Step 3: Differentiate the numerator.
Differentiate the numerator -4x^2 + 8 term by term:
The derivative of -4x^2 is -8x, and the derivative of 8 is 0, so the numerator becomes -8x.

Step 4: Differentiate the denominator.
Differentiate the denominator (x^2 + 2)^2 using the chain rule. Let's denote u = x^2 + 2, then the expression becomes u^2.
Applying the chain rule, differentiate u^2 with respect to u, then multiply by du/dx:

d/dx(u^2) = 2u(du/dx)

The derivative of u = x^2 + 2 is du/dx = 2x.

Therefore, the derivative of the denominator (x^2 + 2)^2 is:
2(x^2 + 2)(2x) = 4x(x^2 + 2).

Step 5: Apply the derivative formula.
Now that you have the differentiated numerator and denominator, plug them into the quotient rule formula:

dy/dx = (-8x * (x^2 + 2) - (-4x^2 + 8) * 4x(x^2 + 2)) / [(x^2 + 2)^2]^2

Step 6: Simplify the expression.
Simplify the numerator and the denominator in the above expression to finalize your answer.

By following these steps, you should be able to find the correct derivative using the quotient rule.