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Lab 01

Derivative Lab

What does the n-th derivative look like without repeated expansion?

Choose a function family, an order up to 12, and a parameter. The lab writes the n-th derivative in closed form, plots it beside the function, compares it with an independent numerical derivative, and checks the book's general integral identity (Ch. 1, eq. 1.5) at a point you choose.

  • Computational Demonstration
  • Classical Foundation
  • Published Book Content
f⁽ⁿ⁾(x) = aⁿ sin(ax + nπ/2)

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Scope of this instrument

Supported

  • Families sin⁡(ax)\sin(ax), cos⁡(ax)\cos(ax), eaxe^{ax}, sinh⁡(ax)\sinh(ax), cosh⁡(ax)\cosh(ax) and 1/(x2+a2)1/(x^2+a^2), with real aa (a≠0a \neq 0 for the rational family).
  • Integer orders 0≤n≤120 \le n \le 12, evaluated for real xx in [−3,3][-3, 3].
  • Phase-shift forms ansin⁡(ax+nπ/2)a^n\sin(ax+n\pi/2) and ancos⁡(ax+nπ/2)a^n\cos(ax+n\pi/2), and the angular form (−1)nn! sin⁡((n+1)φ)/(∣a∣ρn+1)(-1)^n n!\,\sin((n+1)\varphi)/(|a|\rho^{n+1}) for the rational family.
  • An independent numerical derivative from Cauchy's integral formula in the complex plane.
  • The identity dndsn[(F(bs)−F(as))/s]=∫abtnf(n)(ts) dt\frac{d^n}{ds^n}\big[(F(bs)-F(as))/s\big] = \int_a^b t^n f^{(n)}(ts)\,dt for sine and cosine kernels, 0≤n≤80 \le n \le 8, ∣s∣≥0.25|s| \ge 0.25.

Not supported

  • Arbitrary user-entered functions; there is no symbolic differentiation engine.
  • Products, quotients, or compositions beyond the six listed families.
  • Complex parameters, and fractional or negative orders.
  • Proof of any formula: agreement is checked only at the values you choose.

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