{"id":402,"date":"2025-11-11T09:15:00","date_gmt":"2025-11-11T08:15:00","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=402"},"modified":"2025-11-11T12:27:53","modified_gmt":"2025-11-11T11:27:53","slug":"mathbio-aplicaciones-de-las-derivadas-parciales","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=402","title":{"rendered":"MathBio: Aplicaciones de las derivadas parciales"},"content":{"rendered":"<p>Hoy vamos a ver aplicaciones importantes de las derivadas parciales:<\/p>\n<ol>\n<li>Derivaci\u00f3n impl\u00edcita.<\/li>\n<\/ol>\n<h2>Derivaci\u00f3n impl\u00edcita<\/h2>\n<p>Consideremos una ecuaci\u00f3n que define a \\(y\\) en forma impl\u00edcita; es decir, \\(F(x, y)=0\\) es una funci\u00f3n impl\u00edcita en cierta regi\u00f3n de \\(\\mathbb{R}^2\\) entre las variables \\(x\\) e \\(y\\), es f\u00e1cilmente derivable utilizando la derivaci\u00f3n impl\u00edcita: \\[\\frac{\\partial F}{\\partial x}\\frac{dx}{dx}+\\frac{\\partial F}{\\partial y}\\frac{dy}{dx}=0,\\] si \\(\\frac{\\partial F}{\\partial y}\\neq 0\\), tendremos \\[\\frac{dy}{dx}=-\\frac{\\partial_x F}{\\partial_y F },\\]<br \/>\ncon siempre que \\(\\partial_y F\\neq 0\\).<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(x^2-xy+2y^3=0\\), \u00bfcu\u00e1l es el valor de \\(y^\\prime(x_0)\\) en (1,-1)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1a() {\n  var htmlShow1a = document.getElementById(\"html-show1a\");\n  if (htmlShow1a.style.display === \"none\") {\n    htmlShow1a.style.display = \"block\";\n  } else {\n    htmlShow1a.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1a\" style=\"display: none;\">\nConsideremos \\(F(x,y)=x^2-xy+2y^3\\), entonces \\[\\frac{dy}{dx}=-\\frac{\\partial_x F}{\\partial_y F}=-\\frac{2x-y}{6y^2-x}\\]<br \/>\nSustituimos en (1,-1) \\[\\frac{dy}{dx}=-\\frac{3}{5}.\\]\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea \\(f(x,y)=\\frac{x}{(x+y)^2}\\). \u00bfCu\u00e1nto vale \\(\\frac{dy}{dx}(1,0)\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv48() {\n  var htmlShow48 = document.getElementById(\"html-show48\");\n  if (htmlShow48.style.display === \"none\") {\n    htmlShow48.style.display = \"block\";\n  } else {\n    htmlShow48.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv48()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show48\" style=\"display: none;\">\nNos piden el valor de \\(\\frac{dy}{dx}\\) en el punto (1,0). Necesitamos determinar la derivada impl\u00edcita, considerando \\(F(x,y)=\\frac{x}{(x+y)^2}\\):\\[\\frac{dy}{ dx }= -\\frac{\\partial_x F}{\\partial_y F }=-\\frac{\\frac{1}{{{\\left( y+x\\right) }^{2}}}-\\frac{2 x}{{{\\left( y+x\\right) }^{3}}}}{-\\frac{2 x}{{{\\left( y+x\\right) }^{3}}}}=\\frac{y-x}{2 x}.\\]<br \/>\nSolo nos resta sustituir en (1,0): \\[\\frac{dy}{ dx }(1,0)=-\\frac{1}{2}\\]\n<\/div>\n<hr \/>\n<p>En el caso \\(F(x, y, f(x, y)) = 0\\) si \\(z = f(x, y)\\) define una funci\u00f3n impl\u00edcita para \\(z\\) en t\u00e9rminos de \\(x\\), \\(y\\) entonces podemos calcular sus derivadas parciales de la siguiente manera, usando la regla de la cadena<br \/>\n\\[\\frac{\\partial F}{\\partial x }\\ \\frac{\\partial x}{\\partial x }+<br \/>\n\\frac{\\partial F}{\\partial y }\\ \\frac{\\partial y}{\\partial x }+<br \/>\n\\frac{\\partial F}{\\partial z }\\ \\frac{\\partial z}{\\partial x }=0\\Rightarrow \\]<br \/>\n\\[\\frac{\\partial F}{\\partial x }\\ \\frac{\\partial x}{\\partial x }+<br \/>\n\\frac{\\partial F}{\\partial z }\\ \\frac{\\partial z}{\\partial x }=0\\]<br \/>\nya que \\(\\frac{\\partial y}{\\partial x }=0\\). Luego<br \/>\n\\[\\frac{\\partial z}{\\partial x }= -\\frac{\\partial_x F}{\\partial_z F },\\]<br \/>\nsiempre que \\(\\partial_z F\\neq 0\\).<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(f(x,y,z)=x^2+y^2+z^2-3\\), \u00bfcu\u00e1l es el valor de \\(\\frac{\\partial z}{\\partial y}(1,1,1)\\) ?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1a13() {\n  var htmlShow1a13 = document.getElementById(\"html-show1a13\");\n  if (htmlShow1a13.style.display === \"none\") {\n    htmlShow1a13.style.display = \"block\";\n  } else {\n    htmlShow1a13.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1a13()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1a13\" style=\"display: none;\">\nConsideremos la funci\u00f3n \\(F(x,y,z)=x^2+y^2+z^2-3\\), tenemos: \\[\\partial_z F=2z,\\ \\partial_y F=2y.\\] Luego<br \/>\n\\[\\frac{\\partial z}{\\partial y}=\\frac{-2y}{2z}=-\\frac{y}{z}.\\]<br \/>\nAhora sustituimos en el punto (1,1,1)<br \/>\n\\[\\frac{\\partial z}{\\partial y}(1,1,1)=-1\\]\n<\/div>\n<hr \/>\n<h3>Recta tangente a una curva en  \\(\\mathbb {R} ^{2}\\)<\/h3>\n<p>Atendiendo lo que hemos visto anteriormente, si \\(F(x,y)=0\\) representa una curva diferenciable en \\(\\mathbb {R} ^{2}\\), la recta tangente en un punto \\((x_0,y_0)\\in\\mathbb {R} ^{2}\\) de la curva, vendr\u00e1 dada por<br \/>\n\\[[\\partial_xF(x_0,y_0)\\ \\partial_yF(x_0,y_0)]\\begin{bmatrix}  x-x_0\\\\ y-y_0 \\end{bmatrix}=0\\]<br \/>\nque es lo mismo que \\[ \\left ( y-y_0 \\right )=-\\left ( \\frac{{\\partial_x F(x_0,y_0)}}{\\partial_y F(x_0,y_0)} \\right )\\left ( x-x_0 \\right ).\\]<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Determinar la ecuaci\u00f3n de la recta tangente a la curva \\(xy-y^2-2y^3=0\\), en el punto (1,-1) <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv5() {\n  var htmlShow5 = document.getElementById(\"html-show5\");\n  if (htmlShow5.style.display === \"none\") {\n    htmlShow5.style.display = \"block\";\n  } else {\n    htmlShow5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show5\" style=\"display: none;\">\nConsideremos \\(F(x,y)=xy-y^2-2y^3\\). Para calcular la recta tangente necesitamos conocer \\(y'(1)\\), luego tenemos que determinar \\(\\frac{dy}{dx}\\), para ello utilizamos la derivada impl\u00edcita \\[\\frac{dy}{dx}=-\\frac{\\partial_x F}{\\partial_y F }=-\\frac{y}{-6 {{y}^{2}}-2 y+x}.\\]<br \/>\nDe este modo, en (1,-1), es \\(y'(1)=\\frac{-1}{3}\\). Solo nos resta sustituir en la recta tangente \\[y+1=-\\frac{1}{3}(x-1);\\] es decir,\\[y=-\\frac{1}{3}x-\\frac{4}{3}.\\]\n<\/div>\n<p>Observar que si \\(y=f(x)\\); es decir \\(F(x,y)=F(x,f(x))=0\\), es una funci\u00f3n expl\u00edcita, entonces la expresi\u00f3n anterior es la misma que la conocida \\[ \\left ( y-y_0 \\right )=-\\left ( \\frac{{d f}}{dx}(x_0) \\right )\\left ( x-x_0 \\right ).\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(\\mathbf{T}(x)\\) la recta tangente a \\(x^{1\/2} +y^{1\/2} = 4\\) en (4, 4), \u00bfcu\u00e1nto es \\(\\mathbf{T}(-1)\\)?\n<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1w() {\n  var htmlShow1w = document.getElementById(\"html-show1w\");\n  if (htmlShow1w.style.display === \"none\") {\n    htmlShow1w.style.display = \"block\";\n  } else {\n    htmlShow1w.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1w\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"Matem\u00e1tica Aplicada - Espacio tangente. Ej.1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/GGwf8ZudzTo?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(\\mathbf{T}(x)\\) la recta tangente a \\(x^2 +3xy+y^2 = 5\\) en (1, 1), \u00bfcu\u00e1nto es \\(\\mathbf{T}(-1)\\)?\n<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1w2() {\n  var htmlShow1w2 = document.getElementById(\"html-show1w2\");\n  if (htmlShow1w2.style.display === \"none\") {\n    htmlShow1w2.style.display = \"block\";\n  } else {\n    htmlShow1w2.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1w2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1w2\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"Matem\u00e1tica Aplicada - Espacio tangente. Ej.2 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/aPtN7WrDESo?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Cu\u00e1ntos puntos de la curva \\(x^2y^2+xy=2\\) cumplen que la pendiente de la recta tangente es -1.\n<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1w3() {\n  var htmlShow1w3 = document.getElementById(\"html-show1w3\");\n  if (htmlShow1w3.style.display === \"none\") {\n    htmlShow1w3.style.display = \"block\";\n  } else {\n    htmlShow1w3.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1w3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1w3\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"Matem\u00e1tica Aplicada - Espacio tangente. Ej.3 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/cu7ZcDeHacw?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> En la figura se muestra una l\u00e1mpara colocada tres unidades hacia la derecha del eje \\(y\\) y una sombra creada por la regi\u00f3n el\u00edptica \\(x^2+4y^2\\leq 5\\). Si el punto (-5, 0) est\u00e1 en el borde de la sombra, \u00bfqu\u00e9 tan arriba del eje \\(x\\) est\u00e1 colocada la l\u00e1mpara?<img decoding=\"async\" loading=\"lazy\" class=\"aligncenter size-medium wp-image-582\" title=\"int2\" src=\"http:\/\/uploads.jesussoto.es\/esptang01.png\" alt=\"\" width=\"457\" height=\"228\" \/><\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv1w4() {\n  var htmlShow1w4 = document.getElementById(\"html-show1w4\");\n  if (htmlShow1w4.style.display === \"none\") {\n    htmlShow1w4.style.display = \"block\";\n  } else {\n    htmlShow1w4.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1w4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1w4\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"Matem\u00e1tica Aplicada - Espacio tangente. Ej.4 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/DSEaV9CU2ds?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<p>El pr\u00f3ximo d\u00eda veremos<\/p>\n<ol>\n<li>El gradiente.<\/li>\n<li>La derivada direccional.<\/li>\n<\/ol>\n<hr \/>\n<h3>Bibliograf\u00eda<\/h3>\n<ul>\n<li>Cap\u00edtulo 14 del libro <em>C\u00e1lculo de varias variables<\/em>, de James Stewart.<\/li>\n<\/ul>\n<hr \/>\n<table id=\"yzpi\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"3\" bgcolor=\"#999999\">\n<tbody>\n<tr>\n<td width=\"100%\">\n<p><strong>Ejercicio:<\/strong>  Sea \\(x^2+xy+y^3=0\\), \u00bfcu\u00e1l es el valor de \\(y^\\prime(1)\\) en (1,0)?<\/p>\n<div id=\"menu-a\">\n<ul>\n<li>-3<\/li>\n<li>-2<\/li>\n<li>1<\/li>\n<li>Ninguno de ellos<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv() {\n  var htmlShow = document.getElementById(\"html-show\");\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p><strong>B.)<\/strong><\/p>\n<p><iframe loading=\"lazy\" title=\"Matem\u00e1tica Aplicada - Derivada impl\u00edcita. Ej.5 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/zPmjzhEKFbI?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Hoy vamos a ver aplicaciones importantes de las derivadas parciales: Derivaci\u00f3n impl\u00edcita. Derivaci\u00f3n impl\u00edcita Consideremos una ecuaci\u00f3n que define a \\(y\\) en forma impl\u00edcita; es decir, \\(F(x, y)=0\\) es una funci\u00f3n impl\u00edcita&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-402","post","type-post","status-publish","format-standard","hentry","category-mathbio"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/402","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=402"}],"version-history":[{"count":6,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/402\/revisions"}],"predecessor-version":[{"id":466,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/402\/revisions\/466"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=402"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=402"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=402"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}