Question

How do you use the chain rule to differentiate `y=4(x^3+5)^(3/4)`?

Answer 1

`dy/dx=(9x^2)/((x^3+5)^(1/4))`

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)|)))........ (A)`

let `color(blue)(u=x^3+5)rArr(du)/dx=3x^2`

and `y=4color(blue)(u)^(3/4)rArr(dy)/(du)=3color(blue)(u)^(-1/4)`

substitute these values into (A) and convert u to x

`rArrdy/dx=3color(blue)(u)^(-1/4).3x^2=(9x^2)/(x^3+5)^(1/4)`