Question

How do you find the derivative of `-2x(x^2+3)^-2`?

Answer 1

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:

1. 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!