Question

How do you use the chain rule to differentiate `y=(x^2+4x)^(1/2)`?

Answer 1

`dy/dx=(x+2)/(x^+4x)^(1/2)`

Explanation:

differentiate using the `color(blue)"chain rule"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(a/a)|)))`

let `u=x^2+4xrArr(du)/(dx)=2x+4=2(x+2)`

and `y=u^(1/2)rArr(dy)/(du)=1/2u^(-1/2)`

`rArrdy/dx=1/2u^(-1/2).2(x+2)`

and convert u back into terms of x

`rArrdy/dx=(x+2)/(x^2+4x)^(1/2)`