Derivative of y = x^2
So far the slopes have been constant: a flat line gives
0,
and the line y = x
gives 1. Now we meet a curve:
f(x) = x^2
The parabola is gentle near the bottom and steep on the sides — so its slope can't be a
single number any more. It must change as x changes.
Let's find exactly how.
Work out the difference quotient
Feed f(x) = x^2 into the
difference quotient
and expand (x+h)^2 = x^2 + 2xh + h^2:
\frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h
The x^2 terms cancel, and a factor of h
divides out of what's left, leaving 2x + h. Now shrink the step:
as h \to 0 the leftover h vanishes, and
we're left with the slope of the curve itself.
\frac{d}{dx}\bigl[\,x^2\,\bigr] = 2x
Watch the slope double
The derivative 2x says the slope is exactly twice the
x-value. At x = 1 the
tangent has slope 2; at x = 3 it's
6; at x = 0 it's flat. Slide the
point along the parabola and watch the tangent line tip over — its slope reads
2x at every spot.
See it together
The curve f(x) = x^2 on the left; its derivative
f'(x) = 2x — a straight line through the origin with slope
2 — on the right. Notice the derivative is negative where the
parabola falls, zero at the bottom, and positive where it rises.