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Differentiation from first principles, animated
A five-scene guided mini-lesson on y = x²: secant → tangent → the limit → f′(x) = 2x. Next / Back drives the story (← → keys work too).
gradient of PQ
=
f
(1+
h
) −
f
(1)
h
=
(1+
h
)² − 1²
h
=
1 + 2
h
+
h
² − 1
h
=
2
h
+
h
²
h
=
h
(2 +
h
)
h
= 2 +
h
(legal —
h
≠ 0)
as
h
→ 0: gradient → 2 + 0 =
2
— the tangent gradient we watched
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h
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