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Question

If find in terms of y alone.

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Solution

Let, y= cos 1 x.

The first order derivative is obtained by differentiating the function with respect to x.

dy dx = d( cos 1 x ) dx = 1 1 x 2 = ( 1 x 2 ) 1 2

Again differentiate the above function with respect to x.

d dx ( dy dx )= d dx [ 1 ( 1 x 2 ) 1 2 ] d 2 y d x 2 ={ 1( 1 2 ) ( 1 x 2 ) 3 2 }× d dx ( 1 x 2 ) = 1 2 ( 1 x 2 ) 3 ×( 2x ) = x ( 1 x 2 ) 3

Substitute x=cosy in the above function.

d 2 y d x 2 = cosy ( 1 cos 2 y ) 3 d 2 y d x 2 = ( cosy ) ( sin 2 y ) 3 = ( cosy ) sin 3 y = 1 sin 2 y × ( cosy ) siny

Further simplify the above function.

d 2 y d x 2 =( cotycose c 2 y )

Therefore, the value of d 2 y d x 2 in terms of y is ( cotycose c 2 y ).


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