Question

How do you differentiate `arcsin(2x)`?

Answer 1

`2/(sqrt(1-4x^2)`

Explanation:

`•color(white)(x)d/dx(sin^-1x)=1/(sqrt(1-x^2))`

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

`•color(white)(x)d/dx(sin^-1(f(x)))=1/(sqrt(1-(f(x))^2)xxf'(x)`

`rArrd/dx(sin^-1(2x))`

`=1/(sqrt(1-(2x)^2))xxd/dx(2x)`

`=2/(sqrt(1-4x^2))`

Answer 2

The derivative is `=2/sqrt(1-4x^2)`

Explanation:

Let `y=arcsin(2x)`

`siny=2x`

Differentiating with respect to `x`

`dy/dxcosy=2`

`dy/dx=2/cosy`

We know that

`sin^2y+cos^2y=1`

`cos^2y=1-sin^2y=1-4x^2`

`cosy=sqrt(1-4x^2)`

Therefore,

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