Question

What is the derivative of `y = xsinh^-1(x/3)-(sqrt(9+x^2))`?

Answer 1

`sinh^(-1)(x)`

Explanation:

Note that `(sinh^(-1)(x))'=1/sqrt(x^2+1)`
By the product rule and the chain rule we get

`sinh^(-1)(x)+x * 1/sqrt(x^2/9+1) * 1/3-1/2 * (9+x^2)^(-1/2)2x`

Simplifying
`x*1/sqrt(x^2/9+1)*(1/3)=x/3*3x/sqrt(x^2+9)=x/sqrt(x^2+9)`
we get the result `sinh^(-1)(x)`