Why differentiate y=vx implicitly when using substitution

[thomas49th]

Homework Statement

Consider
y = 2a + ax

find dy/dx

dy/dx = a

That is right is it not, as a is treated merly as a constantNow consider this question:

Use the substitution y = vx to transform the equation:

dy/dx = (4x+y)(x+y)/x²

into

x(dv/dx) = (2+v)²

According to the mark scheme they
differentiate dy/dx implicitally
y = vx
dy/dx = x(dv/dx) + v

BUT why have we differentitated implicitally?

Thanks :)

 

[Mark44]

In your first example, a is assumed to be a constant, so dy/dx = a, as you showed.
In your second example, both x and y are variables, and v is some function of x. In the substitution y = vx, when you differentiate the right side with respect to x, you cannot treat v as a constant as you did in the first example, so you have to use the product rule. You are assuming that both y and v are functions of x, so any differentiation has to be done implicitly.