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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.