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 =( −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 ).

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