`tan^(-1)` का `cos^(-1)((1-x^(2))/(1+x^(2)))` के सापेक्ष अवलन गुणांक ज्ञात कीजिये ।
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माना `y_(1)=tan^(-1)((2x)/(1-x^(2)))`
`x=tan theta` रखने पर ,
`y_(1)=tan^(-1)((2tan theta)/(1-tan^(2)theta))`
या `y_(1)=tan^(-1)(tan 2 theta)`
या `y_(1)=2theta`
या `y_(1)2 tan^(-1)x`
या `(dy_(1))/(dx)=(2)/(1+x^(2)) ` …(i)
पुनः माना `y_(2)=cos^(-1)((1-x^(2))/(1+x^(2)))`
`x=tan theta` रखने पर ,
`y_(2)=cos^(-1)((1-tan^(2)theta)/(1+tan^(2)theta))`
`=cos^(-1)(cos 2 theta)`
`=2 theta = 2 tan^(-1)x`
`:. (dy_(2))/(dx)=(2)/(1+x^(2))`
अब , `(dy_(1))/(dy_(2))=((dy_(1))/(dx))/((dy_(2))/(dx))=((2)/(1+x^(2)))/((2)/(1+x^(2)))=1`