Find dy/dx from first principle if y=2x^2+3x

We'll apply delta method to determine the instantaneous
rate of change of y with respect to x.

dy/dx = lim [f(x +
delta x) - f(x)]/delta x, delta x->0

We also can
write:

dy/dx = lim [f(x + h) - f(x)]/h,
h->0

We'll calculate f(x+h) = 2(x+h)^2 +
3(x+h)

We'll raise to square x +
h:

f(x+h) = 2x^2 + 4xh + 2h^2 + 3x +
3h

dy/dx = lim (2x^2 + 4xh + 2h^2 + 3x + 3h - 2x^2 -
3x)/h

We'll eliminate like
terms:

dy/dx = lim (4xh + 2h^2 +
3h)/h

lim (4xh + 2h^2 + 3h)/h = lim (4x + 2h +
3)

We'll substitute h by
0:

lim (4x + 2h + 3) = 4x +
3

dy/dx = 4x +
3

Substituting x by any value, we can
compute the slope of the tangent to the graph of the function, in the chosen value for
x.