Question

How do you find the derivative of `u=(6+2x^2)^3`?

Answer 1

`(du)/(dx)=12x(6+2x^2)^2`

Explanation:

we will need the chain rule

`(du)/(dx)=color(red)((du)/(dt))color(blue)((dt)/(dx))`

`u=(6+2x^2)^3`

`t=6+2x^2=>color(blue)((dt)/(dx)=4x)`

`u=t^3=>color(red)((du)/(dt)=3t^2)`

`(du)/(dx)=color(red)(3t^2)xxcolor(blue)(4x)`

substitute back and tidy up

`(du)/(dx)=12x(6+2x^2)^2`

Answer 2

`(du)/dx=12x(6+2x^2)^2`

Explanation:

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

`"given "u=f(g(x)" then"`

`(du)/dx=f'(g(x))xxg'(x)larrcolor(blue)"chain rule"`

`u=(6+2x^2)^3`

`rArr(du)/dx=3(6+2x^2)^2xxd/dx(6+2x^2)`

`color(white)(rArr(du)/dx)=12x(6+2x^2)^2`