Question

$\large{\red{\bf Question }}$

if $\sf y = x^\frac{1}{2}$

find the value of $\sf\dfrac{dy}{dx}$

Class - 11th

Juniors stay away​

Answer 1

$\textbf{Given:} \: y = {x}^{ \frac{1}{x} }$

$\textbf{Objective →} \: \text{ To \: find} \: \frac{dy}{dx}$

$\sf \colorbox{pink} {Solution:}$

$log \: y \: = \frac{1}{x} \: log \: x$

$\frac{1}{x} , \frac{1}{x} + \frac{ - 1}{ {x}^{2} } \: log \: x = \frac{1}{y} \: \frac{dy}{dx}$

$⇒ \frac{dy}{dx} = \frac{y}{ {x}^{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [1 - log \: x]$

$⇒ \frac{dy}{dx} = \frac{ {x}^{ \frac{1}{x} } }{ {x}^{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [1 - log \: x]$

$\textbf{Hence the Answer is}$

$\frac{dy}{dx} = \frac{ {x}^{ \frac{1}{x} } }{ {x}^{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [1 - log \: x]$

$\\ \\ \\ \\ \\ \\ \sf \colorbox{gold} {\red(ANSWER ᵇʸ ⁿᵃʷᵃᵇ⁰⁰⁰⁸}$

Answer 2

${\huge{\blue\longrightarrow{\texttt{\orange A\red N\green S\pink W\blue E\purple R\red}}}}$

Given

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

To find

$\frac{dy}{dx}$

Solution

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

Differentiating Both Sides

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