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FIND THE DERIVATIVE BY THE LIMIT PROCESS!

f(x) = x³ + x²

lim h->0 (f(x + h) - f(x)) / (x + h - x)


f(x) =
x^3 + x^2

f(x + h) =
(x + h)^3 + (x + h)^2 =
x^3 + 3x^2 * h + 3x * h^2 + h^3 + x^2 + 2hx + h^2

f(x + h) - f(x) =>
x^3 - x^3 + 3x^2 * h + 3x * h^2 + h^3 + x^2 - x^2 + 2hx + h^2 =>
3x^2 * h + 3x * h^2 + h^3 + 2hx + h^2

Divide that by (x + h - x), which reduces to h. So we're dividing by h

3x^2 + 3xh + h^2 + 2x + h

h goes to 0

3x^2 + 3x * 0 + 0^2 + 2x + 0
3x^2 + 2x

The derivative is 3x^2 + 2x