Question

If find in terms of y alone.

Answer 1

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 =( −coty⋅cose c 2 y )

Therefore, the value of d 2 y d x 2 in terms of y is ( −coty⋅cose c 2 y ).