Question

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

Answer 1

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)))`