Mizar Commentary on Walter Rudin's Principles of Mathematical Analysis/Differentiation

The Derivative of a Real Function
5.1 Definition Mizar does not introduce differentiation via $$\phi(t)$$ as Rudin does, but works directly with linear and rest functions (see FDIFF_1:def 2/3). There is no reference associating the Mizar differentiation with the notation of LIMFUNC3, but there are FDIFF_1:12 and FDIFF_2:49. (On a side note, Mizar differentiation is not restricted to intervals $$[a,b]$$.) $$f$$ is differentiable at $$x$$ means f is_differentiable_in x (FDIFF_1:def 4), $$f'(x)$$ is diff(f,x) (FDIFF_1:def 5). $$f$$ is differentiable on $$E$$ means <tt>f is_differentiable_on E</tt> (<tt>FDIFF_1:def 6</tt>), $$f'$$ is <tt>f`|E</tt> (<tt>FDIFF_1:def 7</tt>). See also <tt>FDIFF_1:def 8</tt> and <tt>POLYDIFF:def 1</tt>. One-side differentiation is covered in <tt>FDIFF_3</tt>.

5.2 Theorem is <tt>FDIFF_1:24/25</tt>

5.3 Theorem (a) is <tt>FDIFF_1:13/18</tt> or <tt>POLYDIFF:14</tt>. (b) is <tt>FDIFF_1:16/21</tt> or <tt>POLYDIFF:16</tt>. (c) is <tt>FDIFF_2:14/21</tt>.

5.4 Examples $$c'=0$$ for constant $$c$$ is <tt>FDIFF_1:11/22</tt>, see also <tt>POLYDIFF:11</tt>. $$x'=1$$ is <tt>FDIFF_1:17</tt>, see also <tt>POLYDIFF:12</tt>, and more general $$(ax+b)'=a$$ is <tt>FDIFF_1:23</tt>. Differentiation of polynomials is given by <tt>POLYDIFF:def 5/6</tt> and <tt>POLYDIFF:61</tt>. The notations seems a little bit sophisticated because polynomials have quite some structure behind them in Mizar, see <tt>PRE_POLY</tt> and the <tt>POLYNOM</tt> series.

5.5 Theorem is <tt>FDIFF_2:23</tt>.

5.6 Examples (a) is <tt>FDIFF_5:7</tt>. (b) is <tt>FDIFF_5:17</tt>, although $$f'(0)$$ has been explicitly excluded, no reference for that.

Mean Value Theorems
5.7 Definition No reference.

5.8 Theorem No reference, although the statement is basically proven within the proof of <tt>ROLLE:1</tt>. #TODO Explicit reference that local extrema of differentiable functions have derivation 0.

5.9 Theorem is <tt>ROLLE:5</tt>.

5.10 Theorem is <tt>ROLLE:3</tt>.

5.11 Theorem (a) is <tt>ROLLE:11</tt> or <tt>FDIFF_2:31</tt>. (b) is <tt>ROLLE:7</tt>. (c) is <tt>ROLLE:12</tt> or <tt>FDIFF_2:32</tt>.

The Continuity of Derivatives
5.12 Theorem No reference. #TODO $$f'(a)<\lambda< f'(b)$$ implies the existence of an $$x$$ such that $$f'(x)=\lambda$$.

Corollary No reference.

L'Hospital's Rule
5.13 Theorem in <tt>L_HOSPIT</tt>, especially <tt>L_HOSPIT:10</tt>.

Derivatives of Higher Order
5.14 Definition $$f^{(n)}$$ is <tt>diff(f,E).n</tt> (<tt>TAYLOR_1:def 5</tt>, see also <tt>TAYLOR_1:def 6</tt>), where $$E$$ is the domain on which $$f^{(n)}$$ is defined.

Taylor's Theorem
5.15 Theorem Set $$c=\alpha$$ and $$d=\beta$$, then $$P(t)$$ is <tt>Partial_Sums(Taylor(f,E,c,d)).(n-'1)</tt> (see <tt>SERIES_1:def 1</tt> and <tt>TAYLOR_1:def 7</tt>), where $$E$$ is the domain on which $$f^{(n)}$$ is defined. The theorem is <tt>TAYLOR_1:27</tt> with $$n+1$$ instead of $$n$$.

Differentiation of vector-valued Functions
5.16 Remark (about complex-valued functions) Differentiation of functions from a subset of the reals to the complex are not formalized in Mizar, but definitions for complex differentiation are given by <tt>CFDIFF_1:def 6-9</tt> and <tt>CFDIFF_1:def 12</tt>, see also <tt>CFDIFF_1:22</tt>. Continuity of differentiable complex functions is given by <tt>CFDIFF_1:34/35</tt>. The differentiation rules $$f+g$$ and $$fg$$ are given by <tt>CFDIFF_1:23/28</tt>, <tt>CFDIFF_1:26/31</tt> respectively. No reference for $$f/g$$.

5.16 Remark (about normed spaces) In <tt>NDIFF_1</tt> differentiation is defined between normed linear spaces (see <tt>NDIFF_1:def 6-9</tt>), i.e. the domain doesn't need to be a subset of the real numbers. No reference for differentiability iff the components are differentiable. See also <tt>PDIFF_1</tt>. That differentiability implies continuity is given by <tt>NDIFF_1:44/45</tt>. $$f+g$$ is given by <tt>NDIFF_1:35/39</tt>. No reference for inner product.

5.16 Remark (about vector-valued functions) For definitions see <tt>NDIFF_3:def 3-7</tt>, see also <tt>NDIFF_3:13</tt>. No reference for differentiability iff the components are differentiable. See also <tt>NDIFF_4</tt>. That differentiability implies continuity is given by <tt>NDIFF_3:22/23</tt>. $$f+g$$ is given by <tt>NDIFF_3:14/17</tt>. No reference for inner product.

5.17 Examples No reference. $$e^z$$ is defined in <tt>SIN_COS:def 14</tt>, $$e^{ix}$$ is given by <tt>SIN_COS:25</tt>.

5.18 Examples No reference.

5.19 Theorem No reference.

Exercises
1. is <tt>FDIFF_2:25</tt>. 2. is <tt>FDIFF_2:37/38</tt> or <tt>FDIFF_2:45</tt>. 3. No reference. 4. No reference. 5. No reference. 6. No reference. 7. see <tt>L_HOSPIT:10</tt>. 8. No reference. 9. No reference. 10. No reference. 11. No reference. 12. No reference. 13. No reference. 14. No reference. 15. No reference. 16. No reference. 17. No reference. 18. No reference. 19. No reference. 20. No reference. 21. No reference. 22. No reference. 23. No reference. 24. No reference. 25. No reference. 26. No reference. #TODO Find reference that $$|f(x)|\leq A|f'(x)|$$ implies $$f=0$$. 27. No reference. #TODO Find reference for initial value problem. 28. No reference. #TODO Find reference for initial value problem. 29. No reference.