If y -cos -1 x- then find d 2 y - dx 2 in terms of y alone-

Given, y = cos^ 1x ⇒ x=cos y Differentiating w.r.t. y, we get dx/dy= sin y ⇒ dy/dx= cosec y ...(i) Again , differentiating w.r.t. x, we get d^2y/dx^2= ...

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Byju's Answer

Standard XII

Mathematics

Particular Solution of a Differential Equation

If y =cos -1 ...

Question

$I f y = c o s^{- 1} x, t h e n f i n d \frac{d^{2} y}{d x^{2}} i n t e r m s o f y a l o n e .$

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Solution

Given, y = $c o s^{- 1} x \Rightarrow x = c o s y$

Differentiating w.r.t. y, we get

$\frac{d x}{d y} = - s i n y \Rightarrow \frac{d y}{d x} = - c o s e c y . . . (i) A g a i n, d i f f e r e n t i a t i n g w . r . t . x, w e g e t \frac{d^{2} y}{d x^{2}} = \frac{d}{d x} (- c o s e c y) = - (- c o s e c y c o t y) \frac{d y}{d x} = c o s e c y c o t y (- c o s e c y) = - c o t y . c o s e c^{2} y (f r o m E q . (i))$

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If y=cos−1x, then find d2ydx2 in terms of y alone.

Given, y = cos−1x ⇒ x=cos y Differentiating w.r.t. y, we get dxdy=−sin y⇒ dydx=−cosec y ...(i)Again ,differentiating w.r.t. x,we getd2ydx2=ddx(−cosec y)=−(−cosec y cot y)dydx=cosec y cot y(−cosec y)=−cot y.cosec2y (from Eq. (i))

$I f y = c o s^{- 1} x, t h e n f i n d \frac{d^{2} y}{d x^{2}} i n t e r m s o f y a l o n e .$