Question

Find StartFraction dy Over dx EndFraction for y equals StartRoot u EndRoot and u equals x squared plus 1.

Answer 1

`(dy)/(dx)=\frac{x}{\sqrt {x^2+1}}`

Explanation:

We want to find:

`(dy)/(dx) \\y=\sqrt u\\(d)/(dx)(√(u))`

Given that:

`u=x^2+1`

Applying the chain rule:

`(dy)/(dx)=(dy)/(du)*(du)/(dx)`

Solving dy/du:

`(dy)/(du)=(d)/(du)(\sqrt u)\\(dy)/(du)=(1)/(2\sqrt u)`

Solving du/dx:

`(d)/(dx)(x^2+1) = 2x`

Therefore, dy/dx is determined by:

`(dy)/(dx)=(1)/(2\sqrt u)*2x \\(dy)/(dx)=(x)/(\sqrt u)\\(dy)/(dx)=\frac{x}{\sqrt {x^2+1}}`