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

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Answer 1 · 2015-08-09T19:04:17

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

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)#