Question

How do you differentiate `y=tan^-1(2x^4)`?

Answer 1

`dy/dx=(8x^3)/(1+4x^8)`

Explanation:

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(d/dx(tan^-1x)=1/(1+x^2))color(white)(2/2)|)))`

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

`color(red)(bar(ul(|color(white)(2/2)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(2/2)|)))to(A)`

`"let " u=2x^4rArr(du)/(dx)=8x^3`

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

Substitute into (A) and change u back to x

`dy/dx=1/(1+u^2)xx8x^3=(8x^3)/(1+4x^8)`