FIND THE DERIVATIVE IF

Y= (2X-5)^2 / (X^2+4)^3

To find the derivative of the given function, we will use the quotient rule and the chain rule. Here's how you can go about it step by step:

Step 1: Rewrite the function in a simplified form:
Y = (2X - 5)^2 / (X^2 + 4)^3

Step 2: Expand the numerator and the denominator:
Y = (4X^2 - 20X + 25) / (X^6 + 12X^4 + 48X^2 + 64)

Step 3: Apply the quotient rule:
The quotient rule states that if we have a function F(x) = U(x) / V(x), then the derivative of F(x) is given by:
F'(x) = (U'(x)V(x) - U(x)V'(x)) / V(x)^2

In our case, U(x) = (2X - 5)^2 and V(x) = (X^2 + 4)^3. Let's find the derivatives of U(x) and V(x) separately:

U'(x) = 2(2X - 5) = 4X - 10
V'(x) = 3(X^2 + 4)^2(2X) = 6X(X^2 + 4)^2

Now we can substitute these derivatives into the quotient rule formula:

Y' = ((4X - 10)(X^2 + 4)^3 - (2X - 5)(6X(X^2 + 4)^2)) / (X^2 + 4)^6

Step 4: Simplify the expression:

Y' = (4X^3 + 16X - 10X^2 - 40 - 12X^3 - 48X^2 + 30X + 75) / (X^2 + 4)^6
= (-8X^3 - 58X^2 + 46X + 35) / (X^2 + 4)^6

So, the derivative of Y with respect to X (dY/dX) is (-8X^3 - 58X^2 + 46X + 35) / (X^2 + 4)^6.