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Episode 1 showed a secant line collapsing into a tangent. This is the algebra for that picture.
The slope of a secant is rise over run: the rise is f(x+h) − f(x), the run is h. That fraction is the difference quotient. The derivative is what it approaches as h goes to zero.
Worked on f(x) = x², step by step: expand (x+h)², the x² terms cancel, every term left has an h, divide through, and you're at 2x + h. Now let h go to zero and the h just disappears leaving 2x.
That cancellation is the whole reason 0/0 isn't a dead end. The h divides out before you ever set it to zero.
Check it: the tangent to x² at x = 3 has slope 6, and 2x at x = 3 is 6.
And the debt from episode 6: at the corner of |x|, the same quotient gives −1 from the left and +1 from the right. They disagree, so the limit doesn't exist, so there is no derivative even though the function is perfectly continuous. A corner has no single slope.
Calculus #7 of 113. Full playlist on my channel.
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2026/09/15
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