y=tan(x)y=tan(x)
tanx=sinx/cosxtanx=sinxcosx
y+Deltay=tan(x+Deltax)
tan(x+Deltax)=sin(x+Deltax)/cos(x+Deltax)
y+Deltay-y=tan(x+Deltax)-tan(x)
sin(A+B)=sinAcosB+cosAsinB
cos(A+B)=cosAcosB-sinAsinB
tan(x+Deltax)=(sinxcosDeltax+cosxsinDeltax)/(cosxcosDeltax-sinxsinDeltax)
Deltay=(sinxcosDeltax+cosxsinDeltax)/(cosxcosDeltax-sinxsinDeltax)-sinx/cosx
Deltay=(((cosx(sinxcosDeltax+cosxsinDeltax)-sinx(cosxcosDeltax-sinxsinDeltax)))/(cosx(cosxcosDeltax-sinxsinDeltax)))
=(cosxsinxcosDeltax+cos^2xsinDeltax-sinxcosxcosDeltax+sin^2xsinDeltax)/(cos^2xcosDeltax-cosxsinxsinDeltax)
=(cos^2xsinDeltax+sin^2xsinDeltax)/(cos^2xcosDeltax-cosxsinxsinDeltax)
=((cos^2x+sin^2x)sinDeltax)/(cos^2xcosDeltax-cosxsinxsinDeltax)
Dividing throughout by
cos^2xcosDeltax
=(tanDeltax+tan^2xtanDeltax)/(1-tanxtanDeltax)
Deltay=(1+tan^2x)(tanDeltax)/(1-tanxtanDeltax)
(Deltay)/(Deltax)=1/(Deltax)xx(1+tan^2x)(tanDeltax)/(1-tanxtanDeltax)
1+tan^2x=sec^2x
(Deltay)/(Deltax)=sec^2x xx1/(Deltax)xx(tanDeltax)/(1-tanxtanDeltax)
applying limits as Deltax->0
lim(Deltay)/(Deltax)=lim(sec^2x xx1/(Deltax)xx(tanDeltax)/(1-tanxtanDeltax))
=sec^2x xx lim(tanDeltax)/(Deltax)/(1-tanx xx limtanDeltax)
lim(tanDeltax)/(Deltax)=1
lim(tanDeltax)=0
Thus,
dy/dx=sec^2x xx 1/(1-tanx xx 0)
dy/dx=sec^2x