If Y = X + Tan X, Show that Cos 2 X D 2 Y D X 2 − 2 Y + 2 X = 0 ? - Mathematics

प्रश्न

If y = x + tan x, show that $\cos^{2} x \frac{d^{2} y}{d x^{2}} - 2 y + 2 x = 0$ ?

उत्तर

Here,

$y = x + \tan x$
$Differentiating w . r . t . x, we get$
$\frac{d y}{d x} = 1 + \sec^{2} x$
$Differentiating again w . r . t . x, we get$
$\frac{d^{2} y}{d x^{2}} = 2 \sec^{2} x \tan x$
$Dividing both sides by \sec^{2} x, we get$
$\cos^{2} x \frac{d^{2} y}{d x^{2}} = 2 \tan x$
$\Rightarrow \cos^{2} x \frac{d^{2} y}{d x^{2}} = 2 (y - x) [∵ y = x + \tan x \Rightarrow \tan x = y - x]$
$\Rightarrow \cos^{2} x \frac{d^{2} y}{d x^{2}} - 2 y + 2 x = 0$

Hence proved.

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Simple Problems on Applications of Derivatives

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पाठ 12: Higher Order Derivatives - Exercise 12.1 [पृष्ठ १६]

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आरडी शर्मा Mathematics [English] Class 12

पाठ 12 Higher Order Derivatives
Exercise 12.1 | Q 3 | पृष्ठ १६

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Simple Problems on Applications of Derivatives

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If Y = X + Tan X, Show that Cos 2 X D 2 Y D X 2 − 2 Y + 2 X = 0 ? - Mathematics

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