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.