A combination of the chain rule and product rule will give you the derivative of this function.

Let your function be #f(x) = g(x) xx h(x)#

The derivative is given by #f'(x) = g'(x) xx h(x) + h'(x) xx g(x)#

The derivative of #g(x)# is simple enough: #g'(x) =- 2#

However, we will need the chain rule to differentiate #h(x)# .

#y = u^(-2)#

#u = (x^2 + 3)#

#dy/dx = dy/(du) xx (du)/dx#

#dy/dx = -2/(u^3) xx 2x#

#dy/dx = -(4x)/(x^2 + 3)^3#

Now, let's use the product rule, as mentioned above, to determine the derivative of #f(x)#.

#f'(x) = g'(x) xx h(x) + h'(x) xx g(x)#

#f'(x) = -2(x^2 + 3)^(-2) + (-2x) xx -(4x)/(x^2 + 3)^3#

#f'(x) = -2/(x^2 + 3)^2 + (8x^2)/(x^2 + 3)^3#

#f'(x) = (8x^2)/(x^2 + 3)^3 - 2/(x^2 + 3)^2#

Hence, the derivative of #y = -2x(x^2 + 3)^(-2)# is #y' = (8x^2)/(x^2 + 3)^3 - 2/(x^2 + 3)^2#

**Practice exercises:**

- Differentiate the following functions.

a) #f(x) = -4x(x^3 + 2x)^2#

b) #g(x) = 2x^2sqrt(x^5 - 9)#

c) #h(x) = 4x^3root(3)(5x^2 - 7)#

d) #i(x) = 2x^4(4x^3 - 11x)^-14#

Hopefully this helps, and good luck!