Mathematics - Mathematical Analysis Partial Derivatives by drifter1

View this thread on: hive.blog | peakd.com | ecency.com

mathematics · @drifter1 · Apr 20 '18 (edited)

$23.70

Mathematics - Mathematical Analysis Partial Derivatives

<html>
<p><img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/X2%2BX%2B1.svg/250px-X2%2BX%2B1.svg.png" width="250" height="250"/></p>
<p><a href="https://en.wikipedia.org/wiki/Partial_derivative">Source</a></p>
<h2>Introduction</h2>
<p>Hello it's a me again drifter1!</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Continuing on with our <strong>Mathematical Analysis</strong> series of Mathematics we today will talk about <strong>Partial Derivatives</strong>.</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Those "new" derivatives are based on the "normal" ones and so I highly suggest you reading my post about that <a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-derivatives">here </a>before getting into this post...if you don't know about them already!</p>
<p>So, without further do, let's get started!</p>
<p><br></p>
<h2>What does a derivative describe?</h2>
<p>&nbsp;&nbsp;&nbsp;&nbsp;When having <strong>only one variabe</strong>, the derivative of course describes the <strong>angle</strong>/rotation of a function's curve to the x axis.</p>
<p>So, the derivative of a function f(x) with x describes the angle towards the x-axis.</p>
<p>But, what about functions of more then one variables?</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Well we saw that limits, continuity and more can also be described for such functions, but we don't talked about differentiation yet...</p>
<p>Depending on the notation the <strong>derivative at the position xo</strong> is given by:</p>
<p><img src="http://quicklatex.com/cache3/0a/ql_bcf43ac10efd16fa1fb362e928c9b50a_l3.png"/></p>
<p>using <a href="http://quicklatex.com/cache3/0a/ql_bcf43ac10efd16fa1fb362e928c9b50a_l3.png">quicklatex</a></p>
<p>So, the derivatives are based on an specific quotient limit...</p>
<p><br></p>
<h2>Partial Derivatives</h2>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Because&nbsp;we can clearly see that the "dependence" variable (x) is in the denominator we start thinking about a way of <strong>describing derivatives for one of the independent input variables</strong>. This exactly is what a partial derivative is.</p>
<p>For two&nbsp;variables we define the partial derivative of z = f(x, y) at the point P(x0, y0) with x as:</p>
<p><img src="https://s14.postimg.cc/v2imq57ep/Screenshot_1.jpg" width="40" height="68"/>= <img src="http://quicklatex.com/cache3/bb/ql_9bd3f11b22a3f87fc1a6cb4731e240bb_l3.png"/></p>
<p><a href="https://en.wikipedia.org/wiki/Partial_derivative">Wikipedia </a>and <a href="http://quicklatex.com/cache3/bb/ql_9bd3f11b22a3f87fc1a6cb4731e240bb_l3.png">Quicklatex</a></p>
<p>If this limit exists (is a real number) then the value is called the <strong>partial derivative o f(x, y) with x at the point P(x0, y0)</strong>.</p>
<p>By doing as we did, we actually define a new function f(x, y0) that is dependent only of x.</p>
<p><br></p>
<p>Using the same principle the derivative of f(x, y) with y is:</p>
<p><img src="http://quicklatex.com/cache3/3c/ql_fbee7dd6c6c5f26743f33c4dc881083c_l3.png"/></p>
<p><a href="http://quicklatex.com/cache3/3c/ql_fbee7dd6c6c5f26743f33c4dc881083c_l3.png">Quicklatex</a></p>
<p><br></p>
<p>For example:</p>
<p><img src="http://quicklatex.com/cache3/56/ql_badd1b2fc855f554c9818187499a0a56_l3.png"/></p>
<p><a href="http://quicklatex.com/cache3/56/ql_badd1b2fc855f554c9818187499a0a56_l3.png">quicklatex</a></p>
<p>The Partial derivatives are:</p>
<p><em>f'x = 2x + 3y</em></p>
<p><em>f'y = -2y + 3x</em></p>
<p><br></p>
<blockquote>We can clearly see that the "other" variable is being treated like a constant.</blockquote>
<p><br></p>
<h2>What does the partial derivative describe?</h2>
<p>Well, it again describes the <strong>angle of a curve.</strong>..</p>
<p>This curve will be <strong>different depending on which variable we choose to derivative with</strong>.</p>
<p>For the function f(x, y) at P(x0, y0) we have:</p>
<ul>
  <li><strong>f'x</strong> or the partial derivative with x at P describes the <strong>angle of f(x, y0)</strong></li>
  <li><strong>f'y</strong> or the partial derivative with y at P describe the <strong>angle of f(x0, y)</strong></li>
</ul>
<p>&nbsp;&nbsp;&nbsp;&nbsp;You can see that we find the angle of a new function g(x) or g(y) which means that we leave the 3-dimensions that a 2-variable functions describes to go to 2-dimensions (from plane to curve/line)</p>
<p><br></p>
<p>But, how can this be useful?</p>
<p><br></p>
<h2>Derivative of Vector Functions</h2>
<p>The derivative of a vector function is based on partial derivatives.</p>
<p>Suppose z = f(x, y) is a plane and r(t) = x(t)i + y(t)j a curve (or vector function) on top of it.</p>
<p>We define a <strong>derivative </strong><em><strong>along </strong></em><strong>the curve</strong> as:</p>
<p><em>f'(t) = df/dt = &nbsp;∂f/∂x * dx/dt + ∂f/∂y * dy/dt</em></p>
<p>where these elements are:</p>
<ol>
  <li>the partial derivative of x(t) with x AND</li>
  <li>the partial derivative of y(t) with y</li>
</ol>
<p>But, because the functions are "complex" we have to use the rule of composition.</p>
<p><br></p>
<p>If we had 3-dimensions the derivative would look like this:</p>
<p><em>f'(t) = df/dt = &nbsp;∂f/∂x * dx/dt + ∂f/∂y * dy/dt + ∂f/∂z* dz/dt</em></p>
<p><br></p>
<p>For <strong>example </strong>if <em>r(t) = cost + sint </em>and f(x, y) = x*y then:</p>
<p><em>f'(t) = df/dt = ∂f/∂x * dx/dt + ∂f/∂y * dy/dt = y*(-sint) + x*cost = </em><em><strong>-sin^2t + cos^2t</strong></em></p>
<p><br></p>
<p>Partial Derivatives are also used for things like:</p>
<ul>
  <li>Finding normal vectors to a plane</li>
  <li>Finding tangent planes</li>
</ul>
<p>Both use directional derivatives that we will talk about next time...</p>
<p><br></p>
<p>And this is actually it for today and I hope that you enjoyed it!</p>
<p>Bye!</p>
</html>

👍 zahnspange, hendrikdegrote, knozaki2015, trumpman, b0y2k, romedog, anyx, lafona-miner, steemstem, aizensou, nextgencrypto, yoshiko, pharesim, richman, ozchartart, thecyclist, ben.zimmerman, meerkat, joseph, platinum-blue, steemychicken1, curie, thebluepanda, eveuncovered, themarkymark, warofcraft, ngc, sirwinchester, ssekulji, ruth-girl, hagie, sherlockcupid, anwenbaumeister, razvanelulmarin, dkmathstats, bigdaddy, minimalpris, kushed, dannystravels, kinakomochi, lemouth, thedelegator, derrick829, yougotflagged, toyman, i-gordan, sunshinetraveler, stitchybitch, timbernana, stoner19, steemservices, prettyrose, thisisbenbrick, spg, arbitrarykitten, dream-o, biancajapan, einsteinpotsdam, jerryblanceton, liberosist, shawnamawna, freefuture, mrfelix, dark.horse, and 73 others

`author`	drifter1
`permlink`	mathematics-mathematical-analysis-partial-derivatives
`category`	mathematics
`json_metadata`	{"tags":["mathematics","analysis","derivatives","steemiteducation","steemstem"],"image":["https://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/X2%2BX%2B1.svg/250px-X2%2BX%2B1.svg.png","http://quicklatex.com/cache3/0a/ql_bcf43ac10efd16fa1fb362e928c9b50a_l3.png","https://s14.postimg.cc/v2imq57ep/Screenshot_1.jpg","http://quicklatex.com/cache3/bb/ql_9bd3f11b22a3f87fc1a6cb4731e240bb_l3.png","http://quicklatex.com/cache3/3c/ql_fbee7dd6c6c5f26743f33c4dc881083c_l3.png","http://quicklatex.com/cache3/56/ql_badd1b2fc855f554c9818187499a0a56_l3.png"],"links":["https://en.wikipedia.org/wiki/Partial_derivative","https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-derivatives","http://quicklatex.com/cache3/0a/ql_bcf43ac10efd16fa1fb362e928c9b50a_l3.png","http://quicklatex.com/cache3/bb/ql_9bd3f11b22a3f87fc1a6cb4731e240bb_l3.png","http://quicklatex.com/cache3/3c/ql_fbee7dd6c6c5f26743f33c4dc881083c_l3.png","http://quicklatex.com/cache3/56/ql_badd1b2fc855f554c9818187499a0a56_l3.png"],"app":"steemit/0.1","format":"html"}
`created`	2018-04-16 16:50:24
`last_update`	2018-04-20 22:30:54
`depth`	0
`children`	7
`last_payout`	2018-04-23 16:50:24
`cashout_time`	1969-12-31 23:59:59
`total_payout_value`	18.509 HBD
`curator_payout_value`	5.193 HBD
`pending_payout_value`	0.000 HBD
`promoted`	0.000 HBD
`body_length`	5,764
`author_reputation`	98,202,866,830,354
`root_title`	"Mathematics - Mathematical Analysis Partial Derivatives"
`beneficiaries`	`[]`
`max_accepted_payout`	1,000,000.000 HBD
`percent_hbd`	10,000
`post_id`	50,413,283
`net_rshares`	4,675,905,931,497
`author_curate_reward`	""

properties (23)vote details (137)

voter	rshares	pct
berniesanders	5,030,378,243	2.5%
pharesim	76,708,043,289	0.15%
lafona-miner	140,325,200,983	5%
kushed	18,047,929,297	0.89%
nextgencrypto	98,462,992,544	2.5%
steemychicken1	51,091,936,147	100%
steemservices	10,153,088,173	2.5%
joseph	53,129,779,436	25%
aizensou	99,564,817,820	25%
b0y2k	285,820,016,418	25%
stoner19	10,278,408,562	25%
drifter1	4,842,613,757	100%
richman	74,868,382,422	25%
anwenbaumeister	25,866,089,948	0.89%
razvanelulmarin	25,691,519,128	25%
anyx	203,730,472,716	20%
raymondspeaks	2,858,542,846	25%
knozaki2015	451,220,753,574	25%
shawnamawna	6,925,713,195	25%
liberosist	7,040,093,539	0.89%
hagie	32,348,154,251	25%
ben.zimmerman	60,978,765,752	100%
coinbar	4,919,135,597	25%
ozchartart	73,345,237,588	2.5%
thebluepanda	47,177,647,858	25%
warofcraft	35,706,747,082	10%
yoshiko	84,018,782,060	25%
thisisbenbrick	9,569,853,103	25%
sirwinchester	34,846,525,706	25%
kiddarko	4,981,478,070	5%
lemouth	14,811,917,032	3.5%
einsteinpotsdam	7,594,834,876	25%
dadview	1,245,212,913	3%
zahnspange	566,916,534,296	25%
thecyclist	64,418,829,112	2.5%
steemsquad	2,416,954,415	25%
allesgruen	2,724,753,762	25%
dannystravels	17,649,657,063	25%
jerryblanceton	7,123,458,747	25%
platinum-blue	51,357,717,603	25%
meerkat	59,980,416,508	0.89%
ssekulji	32,889,864,655	25%
timbernana	10,393,337,146	25%
curie	47,245,929,686	0.89%
sherlockcupid	26,955,240,556	25%
vannour	997,071,004	1%
hendrikdegrote	516,027,101,501	0.89%
steemworld	108,365,468	1%
dark.horse	5,106,163,007	25%
steemstem	125,769,324,954	5%
dashfit	114,456,221	0.44%
ejhaasteem	195,347,362	10%
adventureevryday	4,436,897,015	10%
sethroot	61,997,545	0.08%
toyman	13,211,801,148	25%
i-gordan	12,867,340,243	25%
dna-replication	1,261,966,018	5%
yougotflagged	14,068,610,485	2.5%
buzzbeergeek	4,836,323,414	10%
pacokam8	84,089,736	0.22%
bigdaddy	23,246,769,459	25%
kingsmind	4,875,371,816	25%
michelios	417,740,587	0.08%
steemitcitizen	411,827,507	5%
jade56	111,147,482	2.5%
romedog	279,451,281,156	25%
sunshinetraveler	12,749,465,421	25%
eveuncovered	44,101,532,814	25%
markangeltrueman	335,420,318	0.13%
spg	9,359,969,197	25%
freefuture	6,488,062,309	25%
kinakomochi	17,202,806,618	25%
tantawi	74,395,438	0.89%
trumpman	343,933,962,892	99%
samminator	1,949,945,144	5%
dkmathstats	24,145,419,854	100%
derrick829	14,277,251,460	1%
locikll	422,931,636	1.78%
aboutyourbiz	210,930,121	0.89%
howtostartablog	190,335,581	0.04%
sammosk	4,857,602,764	25%
arbitrarykitten	9,164,386,230	25%
cryptokrieg	245,356,662	0.89%
biancajapan	7,709,159,144	25%
slickhustler007	50,357,649	0.44%
ruth-girl	32,732,815,024	20%
ertwro	1,860,481,252	5%
thedelegator	14,362,282,534	2.5%
thinknzombie	1,565,452,296	0.44%
stitchybitch	12,223,121,935	25%
themarkymark	40,082,446,245	25%
suravsingh	59,409,543	5%
misterakpan	102,271,046	0.04%
minimalpris	21,052,870,877	100%
gabox	175,684,997	0.04%
teneiced	4,592,134,615	25%
massivevibration	4,238,071,939	5%
sumayyahsaidso	4,310,556,231	25%
clweeks	60,531,676	0.35%
torico	293,019,613	0.08%
ksolymosi	975,444,898	5%
simplifylife	151,006,178	0.1%
kriptonoob	717,660,057	1%
jordanx2	59,440,383	0.44%
benainouna	599,209,203	100%
prettyrose	9,737,788,723	100%
erodedthoughts	345,106,760	0.04%
enzor	85,444,733	2.5%
bobdos	697,662,939	0.08%
ngc	35,664,977,147	2.5%
pangoli	434,787,430	5%
skycae	241,161,816	0.89%
robotics101	200,093,851	4%
mrfelix	5,722,598,183	100%
zalandir	133,158,482	0.08%
cgbartow	173,150,436	2.5%
whileponderin	113,515,221	5%
bennettitalia	96,283,600	0.04%
sakura1012	123,821,782	5%
wisewoof	109,841,084	0.44%
thetroublenotes	53,950,928	0.5%
scripture-union	4,118,750,095	100%
dexterdev	1,053,252,824	5%
benleemusic	1,348,658,341	0.04%
chimtivers96	61,756,761	0.89%
zipporah	151,811,412	0.17%
positiveninja	112,242,122	0.44%
spederson	91,493,239	4.5%
chillingotter	102,300,994	0.44%
dream-o	8,095,909,962	25%
effofex	1,809,477,426	100%
victorcovrig	69,791,506	1%
linkyourlife	3,590,541,804	25%
xposed	2,178,061,461	10%
recordpool	365,391,788	10%
niouton	128,983,314	0.17%
engmi	76,212,937	0.44%

@benainouna · Apr 16 '18

$0.02

very good remind of derivatives, i will follow your courses to understand more mathematics from you and to not forget what i learnt before, and i will be thankful you add some excersises in your posts. thank you very much :)

👍 drifter1

`author`	benainouna
`permlink`	re-drifter1-mathematics-mathematical-analysis-partial-derivatives-20180416t134110224z
`category`	mathematics
`json_metadata`	{"tags":["mathematics"],"app":"steemit/0.1"}
`created`	2018-04-16 17:41:15
`last_update`	2018-04-16 17:41:15
`depth`	1
`children`	1
`last_payout`	2018-04-23 17:41:15
`cashout_time`	1969-12-31 23:59:59
`total_payout_value`	0.021 HBD
`curator_payout_value`	0.000 HBD
`pending_payout_value`	0.000 HBD
`promoted`	0.000 HBD
`body_length`	224
`author_reputation`	21,025,921,544,449
`root_title`	"Mathematics - Mathematical Analysis Partial Derivatives"
`beneficiaries`	`[]`
`max_accepted_payout`	1,000,000.000 HBD
`percent_hbd`	10,000
`post_id`	50,420,556
`net_rshares`	4,578,471,188
`author_curate_reward`	""

properties (23)vote details (1)

voter	weight	wgt%	rshares	pct	time
drifter1	0 B		4,578,471,188	100%

@drifter1 · Apr 16 '18

$0.02

Sure :)
I'm mostly posting exercises separately and for many things together, just to keep theory and examples better organized...
Today's topic was not so difficult, but for more advanced stuff I of course have more examples or even explain the theory using an characteristic and describing example.
Glad that you enjoy it and remember/learn :D

👍 drifter1

`author`	drifter1
`permlink`	re-benainouna-re-drifter1-mathematics-mathematical-analysis-partial-derivatives-20180416t182538141z
`category`	mathematics
`json_metadata`	{"tags":["mathematics"],"app":"steemit/0.1"}
`created`	2018-04-16 18:25:39
`last_update`	2018-04-16 18:25:39
`depth`	2
`children`	0
`last_payout`	2018-04-23 18:25:39
`cashout_time`	1969-12-31 23:59:59
`total_payout_value`	0.021 HBD
`curator_payout_value`	0.000 HBD
`pending_payout_value`	0.000 HBD
`promoted`	0.000 HBD
`body_length`	345
`author_reputation`	98,202,866,830,354
`root_title`	"Mathematics - Mathematical Analysis Partial Derivatives"
`beneficiaries`	`[]`
`max_accepted_payout`	1,000,000.000 HBD
`percent_hbd`	10,000
`post_id`	50,426,856
`net_rshares`	4,226,281,097
`author_curate_reward`	""

properties (23)vote details (1)

voter	weight	wgt%	rshares	pct	time
drifter1	0 B		4,226,281,097	100%

@prettyrose · Apr 16 '18

$0.14

Just have a thing for general mathematics and derivatives because of my career line: am an accountant. Really love your blog @drifter1. 

I have love for programming,  but it has always seemed to me as one big thing to acquaint with...any help for people like me? [smiles]

👍 prettyrose, mrfelix, drifter1, scripture-union, zeartul

`author`	prettyrose
`permlink`	re-drifter1-mathematics-mathematical-analysis-partial-derivatives-20180416t170644001z
`category`	mathematics
`json_metadata`	{"tags":["mathematics"],"users":["drifter1"],"app":"steemit/0.1"}
`created`	2018-04-16 17:07:06
`last_update`	2018-04-16 17:07:06
`depth`	1
`children`	4
`last_payout`	2018-04-23 17:07:06
`cashout_time`	1969-12-31 23:59:59
`total_payout_value`	0.141 HBD
`curator_payout_value`	0.000 HBD
`pending_payout_value`	0.000 HBD
`promoted`	0.000 HBD
`body_length`	272
`author_reputation`	5,556,941,549,687
`root_title`	"Mathematics - Mathematical Analysis Partial Derivatives"
`beneficiaries`	`[]`
`max_accepted_payout`	1,000,000.000 HBD
`percent_hbd`	10,000
`post_id`	50,415,721
`net_rshares`	27,989,453,976
`author_curate_reward`	""

properties (23)vote details (5)

voter	rshares	pct
drifter1	4,798,589,996	100%
zeartul	3,220,130,881	11%
prettyrose	9,948,335,507	100%
mrfelix	5,819,591,373	100%
scripture-union	4,202,806,219	100%

@drifter1 · Apr 16 '18

$0.12

Thank you for your kind words...
Programming is interesting and you can create programs/code that can do anything you wish (mostly), but because of this diversity it's difficult to find something easy to start with.
I suggest you to start with something like C.
C is an easy language which has a grammar that is used even in more advanced languages like Java for things that are common.
Start by creating something that just prints out, continue with something that takes your input, does some calculations and gives you an output (or just prints it out) and so on...
It's not so difficult to learn programming anymore!
There are many great sources and videos online :)

👍 prettyrose, mrfelix, drifter1, scripture-union

`author`	drifter1
`permlink`	re-prettyrose-re-drifter1-mathematics-mathematical-analysis-partial-derivatives-20180416t171554580z
`category`	mathematics
`json_metadata`	{"tags":["mathematics"],"app":"steemit/0.1"}
`created`	2018-04-16 17:15:54
`last_update`	2018-04-16 17:15:54
`depth`	2
`children`	3
`last_payout`	2018-04-23 17:15:54
`cashout_time`	1969-12-31 23:59:59
`total_payout_value`	0.120 HBD
`curator_payout_value`	0.000 HBD
`pending_payout_value`	0.000 HBD
`promoted`	0.000 HBD
`body_length`	669
`author_reputation`	98,202,866,830,354
`root_title`	"Mathematics - Mathematical Analysis Partial Derivatives"
`beneficiaries`	`[]`
`max_accepted_payout`	1,000,000.000 HBD
`percent_hbd`	10,000
`post_id`	50,416,974
`net_rshares`	23,971,734,104
`author_curate_reward`	""

properties (23)vote details (4)

voter	rshares	pct
drifter1	4,710,542,473	100%
prettyrose	9,579,878,636	100%
mrfelix	5,625,604,994	100%
scripture-union	4,055,708,001	100%

@prettyrose · Apr 16 '18 (edited)

$0.12

Thanks for your timely response. I hope you dont mind sir, because am deviating from the subject matter (derivatives) under which this comment is presented. Is it possible that i can bring myself to a level of relatively great relevance in the programming ecosystem without having any formal (classroom/college based) education in Computer Science? Cos most of the Tech guys i have met in the past always gave me the impression: I cannot be as good...

👍 prettyrose, mrfelix, drifter1, scripture-union

`author`	prettyrose
`permlink`	re-drifter1-re-prettyrose-re-drifter1-mathematics-mathematical-analysis-partial-derivatives-20180416t172317177z
`category`	mathematics
`json_metadata`	{"tags":["mathematics"],"app":"steemit/0.1"}
`created`	2018-04-16 17:23:24
`last_update`	2018-04-16 17:28:54
`depth`	3
`children`	2
`last_payout`	2018-04-23 17:23:24
`cashout_time`	1969-12-31 23:59:59
`total_payout_value`	0.114 HBD
`curator_payout_value`	0.003 HBD
`pending_payout_value`	0.000 HBD
`promoted`	0.000 HBD
`body_length`	451
`author_reputation`	5,556,941,549,687
`root_title`	"Mathematics - Mathematical Analysis Partial Derivatives"
`beneficiaries`	`[]`
`max_accepted_payout`	1,000,000.000 HBD
`percent_hbd`	10,000
`post_id`	50,418,063
`net_rshares`	23,503,783,182
`author_curate_reward`	""

properties (23)vote details (4)

voter	rshares	pct
drifter1	4,666,518,711	100%
prettyrose	9,369,331,853	100%
mrfelix	5,496,280,741	100%
scripture-union	3,971,651,877	100%

2 replies