#y=tan^2 2x=(tan2x)^2#.
#:. dy/dx=(2tan2x)d/dx(tan 2x)............"[Chain Rule]"#,
#=(2tan2x)(sec^2 2x)d/dx(2x)#,
#=(2tan2x)(sec^2 2x)(2)#,
# rArr dy/dx=4(1+tan^2 2x)tan2x#.
Let us note that,
#dy/dx=4(1+y)y^(1/2)=4(y^(1/2)+y^(3/2))............(square)#.
#:. (d^2y)/dx^2=d/dx{dy/dx}=d/dx{4(y^(1/2)+y^(3/2))}#,
#=4d/dx(y^(1/2)+y^(3/2))#,
#=4d/dy(y^(1/2)+y^(3/2))*dy/dx........."[The Chain Rule]"#,
#=4(1/2y^(-1/2)+3/2y^(1/2))dy/dx#,
#=4*1/2y^(-1/2)(1+3y)dy/dx#,
#=2y^(-1/2)(1+3y){4(y^(1/2)+y^(3/2))}......[because, (square)]#.
#=8(1+3y){y^(-1/2)(y^(1/2)+y^(3/2))}#,
#=8(1+3y){1+y}#,
#:. (d^2y)/dx^2=8(1+4y+3y^2)," or what is the same as to say, "#
# (d^2y)/dx^2-24y^2-32y-8=0#.
Feel & Spread the Joy of Maths.!
