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

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Answer 1 · 2018-07-23T20:04:21

$sinh^(-1)(x)$

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 $sinh^(-1)(x)$

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