Pith Derivative

Calculus Level 5

Evaluate

[dπdxπx=ϕ(xe)].\left[ \left.\frac{d^{\pi}}{dx^{\pi}}\right|_{x=\phi}(x^{e}) \right].

Details and Assumptions

  • [x][x] is the nearest integer function.

  • ϕ=1+52.\phi=\frac{1+\sqrt{5}}{2}.

  • e=limn(1+1n)n.e=\displaystyle \lim_{n \rightarrow \infty}{\left ( 1+\frac{1}{n} \right )^n}.

  • π=2011+x2dx.\displaystyle{\pi=2 \int_{0}^{\infty}{\frac{1}{1+x^2}dx}}.

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