Home » Blog » Find the second derivative of y = (arcsin x)/(1-x2)1/2

Find the second derivative of y = (arcsin x)/(1-x2)1/2

If

    \[ y = \frac{\arcsin x}{\sqrt{1-x^2}} \]

compute \frac{d^2 y}{dx^2}.


We compute the first derivative,

    \begin{align*}  \frac{dy}{dx} &= \left( \frac{1}{\sqrt{1-x^2}} \right) \left( \frac{1}{\sqrt{1-x^2}} \right) + (\arcsin x) \left( \frac{x}{(1-x^2)^{\frac{3}{2}}} \right) \\  &= \frac{1}{1-x^2} + \frac{ x \arcsin x}{(1-x^2)^{\frac{3}{2}}}.  \end{align*}

Taking the derivative of this,

    \begin{align*}  \frac{d^2 y}{dx^2} &= \frac{2x}{(1-x^2)^2} + \frac{x}{(1-x^2)^2} + \frac{\arcsin x}{(1-x^2)^{\frac{3}{2}}} + \frac{3x^2 \arcsin x}{(1-x^2)^{\frac{5}{2}}} \\  &= \frac{3x}{(1-x^2)^2} + \frac{(1+2x^2)\arcsin x}{(1-x^2)^{\frac{5}{2}}}. \end{align*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):