y is an (unknown) function of x
Derive the left hand side
d/dx(1) = 0ddx(1)=0
Derive the right hand side
d/dx(tan(x) - y^2) =d/dx(tan(x) - d/dx(y^2)ddx(tan(x)−y2)=ddx(tan(x)−ddx(y2)
Sum rule in differention
tan(x) = sin(x)/cos(x) tan(x)=sin(x)cos(x)
d/dx(tan(x)) = d/dx(sin(x)/cos(x)) ddx(tan(x))=ddx(sin(x)cos(x))
d/dx(tan(x)) =(sin(x)*sin(x) - (-cos(x))*cos(x))/cos(x)^2 ddx(tan(x))=sin(x)⋅sin(x)−(−cos(x))⋅cos(x)cos(x)2
d/dx(tan(x)) =(sin^2(x) +cos^2(x))/cos^2(x)ddx(tan(x))=sin2(x)+cos2(x)cos2(x)
sin^2(x) +cos^2(x) = 1sin2(x)+cos2(x)=1
d/dx(tan(x)) = 1/cos^2(x)ddx(tan(x))=1cos2(x)
d/dx(y^2) = 2*y*d/dx(y) = 2*y*dy/dx ddx(y2)=2⋅y⋅ddx(y)=2⋅y⋅dydx
The chain rule.
d/dx(tan(x)) = 1/cos^2(x) - 2ydy/dx #
Left hand side = right hand side
0 = 1/cos^2(x) - 2*y*dy/dx0=1cos2(x)−2⋅y⋅dydx
Rewrite
2*y*dy/dx = 1/cos^2(x) 2⋅y⋅dydx=1cos2(x)
