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# If Y = X Sin ( a + Y ) ,Prove that D Y D X = Sin 2 ( a + Y ) Sin ( a + Y ) − Y Cos ( a + Y ) ? - CBSE (Arts) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

If $y = x \sin \left( a + y \right)$ ,Prove that $\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}$ ?

#### Solution

$\text{ We have, y } = x \sin\left( a + y \right)$

Differentiate with respect to y,

$\frac{d y}{d x} = \frac{d}{dx}\left[ x \sin\left( a + y \right) \right]$

$\Rightarrow \frac{d y}{d x} = x\frac{d}{dx}\left\{ \sin\left( a + y \right) \right\} + \sin\left( a + y \right)\frac{d}{dx}\left( x \right) ..........\left[\text{using product rule and chain rule} \right]$

$\Rightarrow \frac{d y}{d x} = x \cos\left( a + y \right)\frac{d}{dx}\left( a + y \right) + \sin\left( a + y \right)\left( 1 \right)$

$\Rightarrow \frac{d y}{d x}\left\{ 1 - x \cos\left( a + y \right) \right\} = \sin\left( a + y \right)$

$\Rightarrow \frac{d y}{d x} = \frac{\sin\left( a + y \right)}{1 - x \cos\left( a + y \right)}$

$\Rightarrow \frac{d y}{d x} = \frac{\sin\left( a + y \right)}{1 - \frac{y}{\sin\left( a + y \right)} \cos\left( a + y \right)}\left[ \because y = x\sin\left( a + y \right) \right]$

$\Rightarrow \frac{d y}{d x} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}$

Hence proved

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Solution If Y = X Sin ( a + Y ) ,Prove that D Y D X = Sin 2 ( a + Y ) Sin ( a + Y ) − Y Cos ( a + Y ) ? Concept: Simple Problems on Applications of Derivatives.
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