. Le të supozojmë këtu se koordinatat e këtij vektori janë funksione të parametrit ( paraqet kohën), pra: , dhe . Nga këto
del se , ku me ndryshimin e parametrit , ndryshohet intensiteti dhe
drejtimi i vektorit , prandaj themi se është një funksion vektorial prej argumentit skalar . Derivati i këtij funksioni vektorial sipas parametrit (kohës) përcaktohet me formulën:
. (64)
- S h e m b u l l i 46. - Të vërtetohet se këndi ndërmjet tangjentes së spirales logaritmike
dhe radius vektorit të saj është konstante.
- Z g j i d h j e : Njehsojmë
dhe këtë vlerë zëvendësojmë në formulën (63):
- prej nga marrim :
![{\displaystyle \varphi =\mathrm {arc} \ \mathrm {ctg} \ a=\mathrm {const.} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b838d5e5fa80742e153ed11e9ea1279b45fb1f61)
3.9. TABELA E FORMULAVE DHE RREGULLAVE THEMELORE TË DERIVIMIT
- 1.
![{\displaystyle \left(c\right)'\!=\!0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/901404a9245e4bef48f1a7a306b41c1b79dcbf9e)
- 2.
![{\displaystyle \left(x\right)'\!=\!1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/557fe2fff0cb6f4c53f387f8424ecc5752da9b4a)
- 3.
![{\displaystyle \left(cu\right)'\!=\!cu'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7022ed2fa17f16d541b7525a2b61c24b73c1033b)
- 4.
![{\displaystyle \left(u\!+\!v\!-\!w\right)'\!=\!u'\!+\!v'\!-\!w'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/739319f070944da2dfac904a938c64e636b1b718)
- 5.
![{\displaystyle \left(uv\right)'\!=\!u'v\!+\!uv'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/867d70bec7ae7d5d0a70515589a21bae629f749f)
- 6.
![{\displaystyle \left({\frac {u}{v}}\right)'\!=\!{\frac {u'v\!-\!uv'}{v^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/038e9dde92c569d6d9ac2a85a89f24bec7f32c38)
- 7.
![{\displaystyle y'_{x}\!=\!y'\cdot u'_{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01661ecebf0dd0dffb70459d415cfaf6fcec3632)
- 8.
![{\displaystyle y'_{x}\!=\!{\frac {1}{x'_{y}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e80bd645c4dda9ddc759e7404a4cd0bd2e2e8d5c)
- 9.
![{\displaystyle y'_{x}\!=\!{\frac {y'_{t}(t)}{x'_{t}(t)}}\!=\!{\frac {\dot {y}}{\dot {x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/460f30efa94567ba2c0355756478d588c7e39cd3)
- 10.
![{\displaystyle \left[F\left(x,y\right)\!=\!0\right]'\Rightarrow F'_{x}\!+\!F'_{y}\cdot y'\!=\!0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84ce90cd5c45f71630338e18a54aa7d43ebfbb0e)
- 11.
![{\displaystyle \left(u^{a}\right)'\!=\!au^{a\!-\!1}u'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91a6953e66568b42807249bf28827a389f676c73)
- 12.
![{\displaystyle \left({\frac {c}{u}}\right)'\!=\!\!-\!{\frac {cu'}{u^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78d21daeabcb114084f93a694a081bb57f2f3856)
- 13.
![{\displaystyle \left({\sqrt {u}}\right)'\!=\!{\frac {u'}{2{\sqrt {u}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ad0ff860f27258de2d846a3ae71d8cfda81e2c)
- 14.
![{\displaystyle \left(\log _{a}u\right)'\!=\!{\frac {u'}{u}}\log _{a}e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4028e39388ce6b8252a11c0fd8dac25ee0c81057)
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- 15.
![{\displaystyle \left(\ln u\right)'\!=\!{\frac {u'}{u}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce08219865da010341bab6858746d68d2c9967cd)
- 16.
![{\displaystyle \left(a^{n}\right)'\!=\!a^{n}\ln a\cdot u'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54458c23723b0fd70a65c1aaf8feecd224b361c6)
- 17.
![{\displaystyle \left(e^{n}\right)'\!=\!e^{u}u'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac30c51a1dd21fd4d701d680bf96a942e0e1c0e7)
- 18.
![{\displaystyle \left(\sin u\right)'\!=\!\cos u\cdot u'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e5af2a3fa9761d7241befa9676d3f81e476a3f)
- 19.
![{\displaystyle \left(\cos u\right)'\!=\!\!-\!\sin u\cdot u'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8cc2728f7fe9c4b1792b0b4318d36d3cc250a6e)
- 20.
![{\displaystyle \left(\mathrm {tg} \ u\right)'\!=\!{\frac {u'}{\cos ^{2}u}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dcad8cf0fb5b5e7ab2e804e3557d9299ddc4cb7)
- 21.
![{\displaystyle \left(\mathrm {ctg} \ u\right)'\!=\!\!-\!{\frac {u'}{\sin ^{2}u}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa2546aa4001c7eb16aab467a6b3f49319a96df)
- 22.
![{\displaystyle \left(\mathrm {arc} \ \sin u\right)'\!=\!{\frac {u'}{\sqrt {1\!-\!u^{2}}}},\ |u|<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/907197a095c05cfba3bb9faa180e02e0f76c45be)
- 23.
![{\displaystyle \left(\mathrm {arc} \ \cos u\right)'\!=\!\!-\!{\frac {u'}{\sqrt {1\!-\!u^{2}}}},\ |u|<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2e962807dfb48bb3e168d493a31ef823c21cb0)
- 24.
![{\displaystyle \left(\mathrm {arc\ tg} \ u\right)'\!=\!{\frac {u'}{1\!+\!u^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc99390c74b9253666b151e0ff7586d1545d1664)
- 25.
![{\displaystyle \left(\mathrm {arc\ ctg} \ u\right)'\!=\!\!-\!{\frac {u'}{1\!+\!u^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cccad7e4a3459da8b574d73d579fc1b8841b989e)
- 26.
![{\displaystyle \left(u^{v}\right)'\!=\!vu^{v\!-\!1}u'\!+\!u^{v}v'\ln u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e7b1fa800f4ac97588cef977e1452cad9567398)
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