Question

find d^2y/dx^2 if y=x^2/1+2x​

Answer 1

Step-by-step explanation:

We have,

$y = \frac{ {x}^{2} }{1 + 2x}$

Differentiating both sides w.r.t x, we have,

$\frac{dy}{dx} = \frac{2x(1 + 2x) - 2( {x}^{2}) }{(1 + 2x)^{2} }$

$\implies \frac{dy}{dx} = \frac{2x + 4 {x}^{2} - 2 {x}^{2} }{ {(1 + 2x)}^{2} }$

$\implies \frac{dy}{dx} = \frac{2x+ 2x^{2} }{ {(1 + 2x)}^{2} }$

$\implies \frac{ {d}^{2} y}{d {x}^{2} } = \frac{2(1 + 2x)(1 + 2x)^{2} - 8x(1 + 2x)(1 + x) }{ {(1 + 2x)}^{4} }$

$\implies \frac{ {d}^{2} y}{d {x}^{2} } = \frac{2(1 + 2x)((1 + 2x)^{2} - 4x(1 + x)) }{ {(1 + 2x)}^{4} }$

$\implies \frac{ {d}^{2} y}{d {x}^{2} } = \frac{2(1 + 4 {x}^{2} + 4x - 4x - 4 {x}^{2} ) }{ {(1 + 2x)}^{3} }$

$\implies \frac{ {d}^{2}y }{d {x}^{2} } = \frac{2}{(1 + 2x) ^{3} }$