Question

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

Answer 1

`dy/dx=3/sqrt(1-9x^2)`

Explanation:

An alternative method that doesn't require knowing the derivative of `arcsin(x)`:

`y=arcsin(3x)`

`sin(y)=3x`

Differentiate both sides with respect to `x`. The derivative of `sin(x)` is `cos(x)`, so the derivative of `sin(y)` is `cos(y)dy/dx`. The derivative of `3x` is `3`.

`cos(y)dy/dx=3`

Solving for the derivative, `dy/dx`:

`dy/dx=3/cos(y)`

We know that `sin(y)=3x`, so we can rewrite the function using all `x` terms using the identity `cos(y)=sqrt(1-sin^2(y))`, which comes from the Pythagorean Identity:

`dy/dx=3/sqrt(1-sin^2(y))`

Since `sin(y)=3x`:

`dy/dx=3/sqrt(1-(3x)^2)`

`dy/dx=3/sqrt(1-9x^2)`