निम्न फलनों का x के सापेक्ष अवकल गुणांक ज्ञात कीजिये -
` (i) cot ^(-1) ((x)/(a) ) `
`(ii) tan ^(-1) (2x+1) `
` " (iii) log tan ^(-1) x `
`(iv) tan ^(-1) (mtan x ) `

Explanation

(i ) माना ` y= cot ^(-1) ((x)/(a) ) ` माना ` (x)/(a) =t `
` therefore " " y= cot^(-1) (t) `
` " " therefore (dy)/(dx) =(d)/(dt) cot ^(-1) (t) (d)/(dx) ((x)/a) `
` " "= (-1)/( a+t^(2) )*(1)/(a) =- ( a^(2))/( a^(2) + x^(2)) *(1)/(a ) =(-a)/( a^(2) + x^(2)) `
(ii) माना ` y= tan ^(-1) (2x+ 1) ` माना ` (2x+ )=t `
` therefore y= tttan ^(-1) (t)`
` therefore ( dy)/(dx) =(d)/(dt) tan ^(-1) ""(d)/(dx) (2x+ 1) =(1.2) /(1+t^(2))`
` " "= (2) /(1+( 2x+ 1) ^(2) )`
(iii) माना ` y= log tan ^(-1) x, tan ^(-1) x=t `
` therefore " "(dy)/(dx)=(d)/(dt) log t (d)/(dx) tan ^(-1) x `
` =(1)/(t)* (1) /(1+x^(2)) = (1)/(tan ^(-1)x(1+ x^(2))`
(iv) माना ` y=tan ^(-1) (mtan x) `
` therefore (dy)/(dx) =(d)/(dt) tan ^(-1) (m tan x ) `
` " "= (1)/( 1+ ( m tan x )^(2))(d)/(dx) ( mtan x )`
या ` (dy)/(dx) =( msec^(2)x ) /( 1+ (m tan x )^(2))`