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

1 Answer
Guilherme N.
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)))#

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