Question 1210577
I assume that in your class what was called the first principle of differentiation is that the derivative of a function {{{f(x)}}} is
the limit of {{{(f(x+DELTA(x))-f(x))/DELTA(x)}}} when {{{DELTA(x)}}} tends to zero.
 
For {{{y=x}}} or {{{f(x)=x}}} , {{{f(x+DELTA(x))=x+DELTA(x)}}} , {{{(f(x+DELTA(x))-f(x))/DELTA(x)}}}{{{"="}}}{{{(x+DELTA(x)-x)/DELTA(x)}}}{{{"="}}}{{{DELTA(x)/DELTA(x)=1}}} ,
but {{{1}}} does not depend on {{{DELTA(x)}}} ,
so the limit of {{{1}}} when {{{DELTA(x)}}} tends to zero is {{{1}}} and {{{highlight(df/dx=1)}}} or {{{highlight(dy/dx=1)}}}
 
 
For {{{y=2x^2—x}}} or {{{f(x)=2x^2—x}}} ,
{{{f(x+DELTA(x))=2(x+DELTA(x))^2-(x+DELTA(x)))}}}{{{"="}}}{{{2(x^2+2DELTA(x)*x+(DELTA(x))^2)-x-DELTA(x)}}}{{{"="}}}{{{2x^2+4DELTA(x)*x+2(DELTA(x))^2-x-DELTA(x)}}}{{{"="}}}{{{2x^2+4DELTA(x)*x+2(DELTA(x))^2-x-DELTA(x)}}} ,
{{{f(x+DELTA(x))-f(x))}}}{{{"="}}}{{{2x^2+4DELTA(x)*x+2(DELTA(x))^2-x-DELTA(x)-(2x^2-x)}}}{{{"="}}}{{{2x^2+4DELTA(x)*x+2(DELTA(x))^2-x-DELTA(x)-2x^2+x)}}}{{{"="}}}{{{4DELTA(x)*x+2(DELTA(x))^2-DELTA(x))}}} ,
{{{(f(x+DELTA(x))-f(x))/DELTA(x)}}}{{{"="}}}{{{(4DELTA(x)*x+2(DELTA(x))^2-DELTA(x))/DELTA(x)}}}{{{"="}}}{{{4x+2x+2DELTA(x)-1}}} , and the limit of {{{4x+2DELTA(x)-1}}} when {{{DELTA(x)}}} tends to zero is {{{4x-1}}} , so {{{highlight(dy/dx=4x-1)}}}