Question

What is the derivative of `arcsin(x/3)`?

Answer 1

`dy/dx=1/sqrt(9-x^2)`

Explanation:

If `y=arcsin(x/3)`
Then `siny=x/3`
Differentiating both sides
`cosy dy/dx=1/3`
To calculate `cosy`, use `cos^2y+sin^2y=1`
So `cosy=sqrt(1-x^2/9)=sqrt(9-x^2)/3`
And the answer is
`dy/dx=1/(3cosy)=3/(3sqrt(9-x^2))=1/sqrt(9-x^2`