`tan ^(-1) ((2x)/(1-x^(2))) ` का ` sin ^(-1) ((2x)/(1+x^(2)))` के सापेक्ष अवकल गुणांक ज्ञात कीजिए|

Explanation

माना `y_1 =tan ^(-1) ((2 x)/(1-x^(2)))`
तथा ` y_2 =sin ^(-1) ((2x )/(1+x^(2))) `
` because " "y_1 =tan ^(-1) ((2x)/(1-x^(2)))`
माना ` x = tan theta rArr theta = tan ^(-1) x `
` therefore (2x)/(1-x^(2) )= (2tan theta )/(1-tantheta ) =tan 2theta `
` therefore " "y_1=tan ^(-1) (tan 2theta )=2theta `
` rArr " "y_1 =2tan ^(-1) x `
` therefore " "(dy_1)/(dx) =2 (d)/(dx) tan ^(-1) x = (2)/(1+x^(2) )`
` therefore " "(dy_1)/(dx)= (2)/(1+x^(2))`
तथा ` y_2 = sin ^(-1) ((2x )/(1+x^(2)))`
माना ` x= tan theta rArr theta =tan ^(-1)x`
` therefore " "(2x)/(1+x^(2) )=(2tan theta )/(1+tan ^(2)theta ) =sin 2theta `
` therefore " "y_2 =sin ^(-1) (sin2theta )= 2theta =2tan ^(-1) x `
` rArr " "y_2 =2tan ^(-1) x `
` therefore " "(dy_2)/(dx) =2 (d)/(dx) tan^(-1) x `
` therefore " "(dy_2)/(dx)=(2)/(1+x^(2))`
अतः` (dy_1)/(dy_2) =(y_1 ("का x के सापेक्ष अवकलन")) / (y_2("का x के सापेक्ष अवकलन"))`
` =(2)/((1+x^(2))/(2/(1+x^(2))))=1 `
` therefore " "(dy_1)/(dy_2)=1 `