Each card is a single, complete idea: what it is, the formulas you need, and one worked example. Built to be re-read.
The average rate of change over [a,b] is the slope of the secant line joining the two endpoints of the interval. It answers 'how fast on average,' not 'how fast right now.'
The instantaneous rate of change is the slope of the tangent line at a single point. You get it exactly by taking a limit of secant slopes as the gap h shrinks to zero.
Instead of computing a slope one point at a time, differentiate once to get a derivative function that returns the instantaneous rate of change at any x. All notations mean the same thing.
A tangent line uses point-slope form with the derivative as the slope. Find the point, find f' at that point, then substitute.
A function is differentiable where it has a single well-defined, smooth slope. It fails at discontinuities, corners/cusps, and vertical tangents. Differentiable ⟹ continuous, but not the reverse.
These shortcuts replace the limit definition. Constants die, constant factors ride along, and sums differentiate term by term.
Bring the exponent down as a coefficient and drop it by one. It works for negative and fractional exponents too — rewrite radicals and fractions as powers first.
For a product of two functions, differentiate one factor at a time and add. The derivative of a product is never just the product of the derivatives.
For a quotient, it is 'low d-high minus high d-low, over low squared.' Order matters here, unlike the product rule.
Memorize the six trig derivatives. The co-functions (cos, cot, csc) all pick up a negative sign.
ex is its own derivative; lnx gives x1. Other bases carry an extra factor of ln(a).
The derivative is the limit of the average rate of change. Differentiability implies continuity, but not the reverse.