When Derivatives Don't Exist

Calculus 1 · Axiom Academy

LESSON When Derivatives Don't Exist The four ways a function fails to have a derivative at a point — corner, cusp, vertical tangent, and discontinuity — each shown as the tangent line breaking down. The classic corner is f(x) = |x| at x = 0 . The graph is two straight rays meeting at a point. The left ray has slope -1 ; the right ray has slope +1 . Each one-sided derivative exists on its own — but they disagree, so there is no single tangent line at the corner. The secant slope holds steady at -1 : the left-hand derivative is -1 . The secant slope holds steady at +1 : the right-hand derivative is +1 . 2. Cusp — the slopes run off to opposite infinities A cusp is a sharper corner: instead of two finite slopes, both one-sided slopes blow up — and in opposite directions. Take f(x) = x^ 2/3 at x = 0 . Coming in from the left the tangent steepens toward straight-down ( ); from the right it steepens toward straight-up ( ). The curve closes to a single sharp point. Slope — the tangent rotates toward vertical, pointing down. Slope — the tangent rotates toward vertical, pointing up. 3. Vertical tangent — infinite slope, but agreeing Sometimes both sides agree on the tangent line and it's still a problem — because that line is vertical. For at x = 0 , a secant through two nearby points stands up straighter and straighter as the points close in. The tangent exists as a vertical line, but its slope is — not a finite number — so f'(0) is undefined.

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