NapkinCalc

Calculus 1 — Limits & Derivatives

The derivative — a limit of slopes

continues from lesson 1 — values defined earlier in the course stay live here

Average speed over an interval is easy: Δf/Δx. The derivative is what happens when the interval shrinks to nothing — the instantaneous slope. Watch the average slope of f(x) = x² at x = 3 converge as h shrinks:

f(x)=x2f\left(x\right) = x^{2}
s1=f(3+1)−f(3)1s_{1} = \frac{\mathrm{f}\left(3 + 1\right) - \mathrm{f}\left(3\right)}{1} = 77 slope over a wide step: 7
s2=f(3+0.01)−f(3)0.01s_{2} = \frac{\mathrm{f}\left(3 + 0.01\right) - \mathrm{f}\left(3\right)}{0.01} = 6.01006.0100 narrower: 6.01
s3=f(3+10−8)−f(3)10−8s_{3} = \frac{\mathrm{f}\left(3 + 10^{-8}\right) - \mathrm{f}\left(3\right)}{10^{-8}} = 6.00006.0000 the derivative: 6
✓ pass abs(s3−6)<0.0001\mathrm{abs}\left(s_{3} - 6\right) < 0.0001 f′(3) = 2·3 — the power rule in the flesh

The pattern you just watched IS the power rule: the derivative of xⁿ is n·xⁿ⁻¹. The engine can do it symbolically too:

diff(x3+2⋅x,x)\mathrm{diff}\left(x^{3} + 2 \cdot x, x\right) = 3⋅x2+23 \cdot x^{2}+2 symbolic: 3x² + 2