Question

is question ke sath answer bhi hai, I just want to know that the formula of Differentiation is
(du/dx)v + (dv/dx)u
is question me dlnx/dx ki sin se multiply kyu nhi hua
please don't scam ​

Answer 1

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Question :

$\\ \large\rm{\dfrac{d}{dx}sin(lnx)}$

To find :

$\\ \large\rm{ Differentiation \: of \: \dfrac{d}{dx}sin(lnx)}$

Explanation:

$\bigstar \pink{given} \: \implies \large \sf \: y = sin(lnx)$

$\\ \implies \bf{ \: differentiating \: both \: \: side \: with \: respect \: to \: x}$

$\\ \implies \large \sf \: y = sin(lnx)$

$\\ \implies \large \sf \: \frac{dy}{dx} = \frac{d}{dx} sin(lnx)$

$\: \: \: \: \: \\ \therefore \bf{by \: using \: chain \: rule}$

$\\ \implies \displaystyle \sf \: \frac{dy}{dx} = \frac{d}{dx} sin(lnx) \times \frac{d}{dx} lnx \: \times \frac{d}{dx} \: x$

Values :

$\bullet \bf \large\ \: \frac{d}{dx} lnx = \frac{1}{x}$

$\bullet \large \bf \: \frac{d}{dx} x = 1$

$\\ \therefore\bf \: put \: the \: values \: in \: given \: expression$

$\\ \implies \large \sf \: \frac{dy}{dx} = \frac{d}{dx} sin(lnx) \times \frac{d}{dx} lnx \times \frac{d}{dx} x$

$\\ \implies \large \sf \: \frac{dy}{dx} = cos(lnx) \times \frac{1}{x} \times 1$

$\\ \implies \large \sf \: \frac{dy}{dx} = \frac{cos(lnx)}{x}$

$\\ \therefore \large \boxed{ \sf{ \frac{dy}{dx} = sin(lnx) = \frac{cos(lnx)}{x} }}$

hence proved it ⛄