If y = - 1 x, find y d x 2 in terms of y alone.

y = - 1 x x = y …(i) Differentiating (i) w.r.t. x, we get 1 = - y dx dx = - ecy …(ii) Differentiating (ii) w.r.t. x, we get y d x 2 = ecy y dx = - e c 2 y y

Continuity and Differentiability — Class 12 Maths Solution

ncert exercise SA NCERT Ex.5.7 ,Q.12,Page 184

Question

If $y = {\cos ^{ - 1}}x,$ find $\cfrac{{{d^2}y}}{{d{x^2}}}$in terms of y alone.

Step-by-step Solution

$y = {\cos ^{ - 1}}x \Rightarrow$ $x = \cos y$ …(i)
Differentiating (i) w.r.t. x, we get
$1 = - \sin y\cfrac{{dy}}{{dx}}$

$\Rightarrow$ $\cfrac{{dy}}{{dx}} = - \cos ecy$ …(ii)

Differentiating (ii) w.r.t. x, we get

$\cfrac{{{d^2}y}}{{d{x^2}}} = \cos ecy\cot y\cfrac{{dy}}{{dx}} = - \cos e{c^2}y\cot y$

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