if y=e^ x, then ( ^2x)y_2= - Vidyadip

MCQ

if $\text{y}=\text{e}^{{\tan}}\ \text{x},$ then $(\cos^2\text{x})\text{y}_2=$

A
$(1-\sin2\text{x})\text{y}_1$
B
$-(1+\sin2\text{x})\text{y}_1$
✓
$(1+\sin2\text{x})\text{y}_1$
D
None of these

✓

Answer

Correct option: C.

$(1+\sin2\text{x})\text{y}_1$

$\text{y}=\text{e}^{{\tan}}\text{x},$
$\text{y}_1=\text{sec}^2\text{xe}^{\tan}\text{x}$
$\Rightarrow\cos^2\text{xy}_1=\text{e}^{\tan}\text{x}$
again differentiating $\text{ w.r.t. x},$ we get
$\cos^2\text{xy}_2-2\cos\text{x}\sin\text{xy}_1=\sec^2\text{xe}^{\tan}\text{x}$
$\Rightarrow\cos^2\text{xy}_2=\text{y}_1\sin2\text{x}+\text{y}_1$

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