Question

Find the derivative

Answer 1

First use the chain rule; take
`y=(x+5)/(x^2+3)`. Then

`(\mathrm df)/(\mathrm dx)=(\mathrm df)/(\mathrm dy)\cdot(\mathrm dy)/(\mathrm dx)`

By the power rule,

`f(x)=y^2\implies(\mathrm df)/(\mathrm dy)=2y=(2(x+5))/(x^2+3)`

By the quotient rule,

`y=(x+5)/(x^2+3)\implies(\mathrm dy)/(\mathrm dx)=((x^2+3)(\mathrm d(x+5))/(\mathrm dx)-(x+5)(\mathrm d(x^2+3))/(\mathrm dx))/((x^2+3)^2)`

`\implies(\mathrm dy)/(\mathrm dx)=((x^2+3)-(x+5)(2x))/((x^2+3)^2)`

`\implies(\mathrm dy)/(\mathrm dx)=(3-10x-x^2)/((x^2+3)^2)`

So

`(\mathrm df)/(\mathrm dx)=(2(x+5))/(x^2+3)\cdot(3-10x-x^2)/((x^2+3)^2)`

`\implies(\mathrm df)/(\mathrm dx)=(2(x+5)(3-10x-x^2))/((x^2+3)^3)`