If y=a x+b x (where a and b are constants), then y^2+ ( d y ~d x )^2 is

Step-by-step solution

y= a x+ b x Differentiating w.r.t. x , we get d y ~d x = a x- b x Now, aligned & y^2+ ( d y ~d x )^2 = & ( a x+ b x)^2+( a x- b x)^2 = & a ^2 ^2 x+ b ^2 ^2 x+2 ab x x+ a ^2 ^2 x & + b ^2 ^2 x-2 ab x x = & a ^2 ( ^2 x+ ^2 x )+ b ^2 ( ^2 x+ ^2 x ) = & a ^2+ b ^2 & y^2+ ( d y ~d x )^2= a ^2+ b ^2 aligned Here, a , b are constants. y^2+ ( d y ~d x )^2 is also a constant.