Exactly one function equals its own derivative, and eˣ is that function: its slope at every point is the value of the function there, which is why it keeps multiplying itself. For every other function the slope must be taken as a limit, as the secant through two points separated by h swings into the tangent while h goes to 0. Running the other way is the integral, the area of n rectangles under the curve converging as n goes to infinity. The Fundamental Theorem of Calculus welds the two directions together: the integral of f from a to b equals F(b) − F(a) whenever F′ = f.
Exactly one function equals its own derivative, and eˣ is that function: its slope at every point is the value of the function there, which is why it keeps multiplying itself. For every other function the slope must be taken as a limit, as the secant through two points separated by h swings into the tangent while h goes to 0. Running the other way is the integral, the area of n rectangles under the curve converging as n goes to infinity. The Fundamental Theorem of Calculus welds the two directions together: the integral of f from a to b equals F(b) − F(a) whenever F′ = f.
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