Higher Derivatives

Real Analysis · Axiom Academy

Differentiate again, and again: how the second derivative reads concavity and acceleration, and what happens when you never stop. 1. Differentiate, Then Differentiate Again If f is differentiable and f' is also differentiable, the second derivative is the derivative of f' . It measures the rate of change of the rate of change — how the slope itself is moving. Three notations for the same object Each derivative knocks the exponent down by one and multiplies by it. After the 4th derivative we hit the constant 24 = 4! , and every derivative beyond that is 0 . 2. Position, Velocity, Acceleration Let s(t) be an object's position at time t . Its first derivative is velocity — how fast position changes. Its second derivative is acceleration — how fast velocity changes. They are one chain of differentiation. Where the object is right now. The slope of position — its rate of change. The slope of velocity — the second derivative of position. At every instant the three are locked together by differentiation. The second derivative is the constant -g : a steady downward acceleration due to gravity, no matter the instant. 3. The Sign of f'' Is Concavity The second derivative tells you which way a graph curves . Where the slope is increasing and the curve bends upward like a ; where the slope is decreasing and it bends downward like a . Watch the curve recolor as the sign flips. 4. Going Higher: Who Survives Forever

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