Question

How do you use the Product Rule to find the derivative of `y = 1/(1-x^2)`?

Answer 1

`y^' = (2x)/(1-x^2)^2`

Explanation:

In order to differentiate this function by using the product rule, you need to find a way to write it as a product of two functions.

Notice that you can write the function as

`1/(1-x^2) = (1-x^2)^(-1) = 1 * (1-x^2)^(-1)`

You can now use the formula

`color(blue)(d/dx(y) = [d/dx(f(x))] * g(x) + f(x) * d/dx(g(x)))`

In your case, you have `f(x) = 1` a nd `g(x) = (1-x^2)^(-1)`.

This means that the derivative of `y` will be

`d/dx(y) = [d/dx(1)] * (1-x^2)^(-1) + 1 * d/dx(1-x^2)^(-1)`

Now, to differentiate `(1-x^2)^(-1)` you can use the chain rule for `u^(-1)`, with `u = 1-x^2`. This will get you

`d/dx(u^(-1)) = d/(du)u^(-1) * d/dx(u)`

`d/dx(u^(-1)) = -u^(-2) * d/dx(1-x^2)`

`d/dx(u^(-1)) = -u^(-2) * (-2x)`

`d/dx(1-x^2)^(-1) = -(1-x^2)^(-2) * (-2x)`

`d/dx(1-x^2)^(-1) = 2x * (1-x^2)^(-2)`

Your target derivative will thus be

`y^' = 0 * (1-x^2)^(-1) + 1 * 2x * (1-x^2)^(-2)`

`y^' = color(green)((2x)/(1-x^2)^2)`