How do you differentiate f(x)= x/sqrt(1-x^2) using the quotient rule?

1 Answer
Dec 30, 2015

We can see this as a product: f(x)=x*(1-x^2)^(-1/2) and use the product rule, as well as - for the second term - the chain rule.

Explanation:

  • Chain rule states that for y=f(x)g(x), then (dy)/(dx)=f'(x)g(x)+f(x)g'(x)
  • Chain rule states that (dy)/(dx)=(dy)/(du)(du)/(dx)

Renaming u=1-x^2, we get that the second term becomes u^(-1/2) and chain rule turns out to be applicable.

(dy)/(dx)=1*(1-x^2)^(-1/2)+x(x/(u^(3/2)))

(dy)/(dx)=1/(1-x^2)^(1/2)+x^2/((1-x^2)^(3/2))

(dy)/(dx)=(1-cancel(x^2)+cancel(x^2))/(1-x^2)^(3/2)=1/(1-x^2)^(3/2)=(1-x^2)^(-3/2)