How do you differentiate f(x) = sqrt[ (3 x + 1) / (5 x^2 + 1) using the chain rule?

1 Answer
Jan 7, 2016

We'll need chain rule here as well, besides the quotient rule itself.

Explanation:

  • Chain rule: (dy)/(dx)=(dy)/(du)(du)/(dx)

  • Quotient rule: (a/b)'=(a'b-ab')/b^2

Renaming u=(3x+1)/(5x^2+1), we have f(x)=sqrt(u). Let's do it:

(dy)/(dx)=1/(2u^(1/2))(3(5x^2+1)-(10x(3x+1)))/(5x^2+1)^2

(dy)/(dx)=1/(2u^(1/2))(15x^2+3-30x^2-10x)/(5x^2+1)^2

(dy)/(dx)=(-15x^2-10x+3)/(2(5x^2+1)^2(sqrt((3x+1)/(5x^2+1)))