DIFFERENTIAL COEFFICIENTS. EXAMPLES. 335 



EXAMPLES. 



L. In the ellipse -r- = ./ f . t _ g j ; and if x = a sin <f>, 



ds 



WfO / /^ O O * \ 



-T7 = a v(l e sm <) 



2. In the parabola y 2 =4aa;, -7- = . /[ -- ) . 



dx V \ x J 



3. In the circle -7- = - . 



dx y 



4. Find the differential coefficient of the arc of the curve 



e"(e x -l)=e* + l. 



ds e** + l 



. -r- = -r 



dx e" - 1 



T ,1 % z z ds a* 



o. In the curve x* + y* = a ? . -7- = r . 



' dc a;4 



6. In the curve r = a (1 + cos 6), -^ = 2a cos - . 



Cits 



ds 



7. In the curve r=o e , -^ = r ^{1 + (log a) 2 ]. 



7 J 



8. In the curve r 2 =a 2 cos20, - = L. 



atf T 



T ,, D ds 



9. In the curve r av, -y- = 



-y- - 

 ar a 



10. If e"" = cos or, -r- = cos x. 

 as 



