Question

Differentiate the following function with respect to x

$y = x + \frac{1}{x + \frac{1}{x + \frac{1}{x} + - - - \infty }}$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\rm \: y = x + \dfrac{1}{x + \dfrac{1}{x + \dfrac{1}{x + \dfrac{1}{x} + - - - \infty } } }$

can be rewritten as

$\rm \: y = x + \dfrac{1}{y}$

$\rm \: y = \dfrac{xy + 1}{y}$

$\rm \: {y}^{2} = xy + 1$

On differentiating both sides w. r. t. x, we get

$\rm \: \dfrac{d}{dx} {y}^{2} = \dfrac{d}{dx}(xy + 1)$

We know,

$\boxed{\tt{ \: \dfrac{d}{dx} {x}^{n} = {nx}^{n - 1} \: }}$

So, using this result, we get

$\rm \: 2y\dfrac{dy}{dx} = \dfrac{d}{dx}(xy) + \dfrac{d}{dx}1$

We know,

$\boxed{\tt{ \: \dfrac{d}{dx}uv \: = \: u\dfrac{d}{dx}v \: + \: v\dfrac{d}{dx}u \: }}$

and

$\boxed{\tt{ \: \dfrac{d}{dx}k \: = \: 0 \: }}$

So, using this result, we get

$\rm \: 2y\dfrac{dy}{dx} = x\dfrac{d}{dx}y + y\dfrac{d}{dx}x + 0$

$\rm \: 2y\dfrac{dy}{dx} = x\dfrac{dy}{dx}+ y \times 1$

$\rm \: 2y\dfrac{dy}{dx} - x\dfrac{dy}{dx} = y$

$\rm \: \dfrac{dy}{dx}(2y- x )= y$

$\bf\implies \:\dfrac{dy}{dx} = \dfrac{y}{2y - x}$

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ADDITIONAL INFORMATION

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}$

Answer 2

$\underline{ \pink{ \rm GIVEN \: \: DATA\: \: : }}$

Function is

$\large\bf y = x + \frac{1}{x + \frac{1}{x + \frac{1}{x} \: . \: . \: . \: \infty } }$

$\underline{ \purple{ \rm TO \: \: FIND \: \: : }}$

Differentiating the Function with respect to x

$\underline{ \red{ \mathfrak{ FINAL \: \: ANSWER\: \: : }}}$

$\rm \frac{dy}{dx} = \frac{y}{2y - x}$

For Clear Explanation, refer the given attachment