निम्न प्रश्नो के प्रचलिक रूप में दिये फलनों में से प्रत्येक के लिए `(dy)/(dx)` ज्ञात कीजिए :
`x=e^theta(theta+1/theta), y= e^(-theta)(theta-1/theta)`

Explanation

प्रश्न से, `x=e^(theta)(theta+(1)/(theta)),y=e^(-theta)(theta-(1)/(theta))`
`therefore(dx)/(d theta)=e^(theta)(1-(1)/(theta^(2)))+(theta+(1)/(theta))e^(theta)`
`=e^(theta)(1-(1)/(theta^(2))+theta+(1)/(theta))=(etheta(theta^(2)-1+theta^(3)+theta))/(theta^(2))`
तथा `(dy)/(d theta)=e^(-0)(1+(1)/(theta^(2)))+(theta-(1)/(theta))(-e^(-theta))`
`=e^(-theta)(1+(1)/(theta^(2))-theta+(1)/(theta))=({e^(-theta)(theta^(2)+1-theta^(3)+theta)})/(theta^(2))`
`therefore(dy)/(dx)=(dy)/(d theta)//(dx)/(d theta)=({e^(-theta)(theta^(2)+1-theta^(3)+theta)})/(theta^(2))xx(theta^(2))/({e^(theta(theta^(2)-1+theta^(3)+theta))})`
`=(e^(-2theta)(theta^(2)+1-theta^(3)+theta))/((theta^(2)-1+theta^(3)+theta))`.