Question

y=x/√1-x2 find d3y/dy3

Answer 1

$y = \frac{x}{ \sqrt{1 - {x}^{2} } } \\ \frac{dy}{dx} = \frac{ \sqrt{1 - {x}^{2} } - \frac{1}{2 \sqrt{1 - {x}^{2} } } \times 2x}{1 - {x}^{2} } \\ \frac{dy}{dx } = \frac{1 - 2 {x}^{2} }{ {(1 - {x}^{2}) }^{ \frac{3}{2} } } \\ \frac{ {d}^{2}y }{d {x}^{2} } = \frac{ - 4x {(1 - {x}^{2}) }^{ \frac{3}{2} } - \frac{3}{2} {(1 - {x}^{2}) }^{2} \times 2x }{( {1 - ({x)}^{2} )}^{3} } \\ \frac{ {d}^{2}y }{d {x}^{2} } = \frac{ - 4x {(1 - {x}^{2}) }^{ \frac{3}{2} } - 3 {(1 - {x}^{2} )}^{ \frac{1}{2} } }{ {(1 - {x}^{2} )}^{3} } \\ \frac{ {d}^{3} y}{d {x}^{3} } = \frac{( ( - 4 {(1 - {x}^{2} )}^{ \frac{3}{2} } - 4 {x}^{2} \sqrt{1 - {x}^{2} } - \frac{3x}{2} {(1 - {x)}^{ \frac{ - 1}{2} } }^{2} ))( {1 - {x}^{2}) }^{3} - ( - 4 ({1 - {x}^{2}) }^{ \frac{3}{2} } - 3 {(1 - {x}^{2}) }^{ \frac{1}{2} }) \times 6x ({1 - {x}^{2}) }^{2} }{ {(1 - {x}^{2} )}^{6} }$
hope it will help you.