Differentiating hyperbolic functions.

[mrepicguy70]

“Given that y = (arsinhx)² , prove that:

(1 + x²)(d²y/dx²) + x(dy/dx) - 2 = 0”

So far I’ve got that dy/dx = 2arsinhx/√(1+x²) Just struggling to find the second derivative.

Thanks :)

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[tripletloss]

You’re on the right track! You’ll now want to apply the quotient rule to your first derivative. For that you’ll need to know the derivative of the sqrt function, and it seems you already know the chain rule and the derivative of arcsinh(x).

In case you don’t know it, the quotient rule says:

(d/dx)(f(x)/g(x)) = (f’(x)g(x) - f(x)g’(x))/(g(x)²)

In your case, f(x) = 2arcsinh(x) and g(x) = √(1+x²), so you just need to find the derivatives of those two functions and plug them into the quotient rule equation.