{"id":6712,"date":"2021-07-21T00:03:09","date_gmt":"2021-07-20T15:03:09","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6712"},"modified":"2021-09-24T00:01:47","modified_gmt":"2021-09-23T15:01:47","slug":"definition-of-a-derivative","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/definition-of-a-derivative\/","title":{"rendered":"\ubbf8\ubd84\uc758 \uc815\uc758"},"content":{"rendered":"<div style=\"display: none; visibility: hidden;\">\n<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<p><!--\n\\[\n\\newcommand{\\vecf}{{\\mathbf{f}}}\n\\newcommand{\\vecL}{{\\mathbf{L}}}\n\\newcommand{\\vecR}{{\\mathbb{R}}}\n\\newcommand{\\imI}{\\boldsymbol{i}}\n\\]\n-->\n<\/div>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 5\uc7a5 1\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\ud568\uc218 \\(y=f(x)\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \\(x\\)\uc640 \\(y\\)\ub294 \uc2e4\uc218 \ubcc0\uc218\uc774\ub2e4. \uadf8\ub9ac\uace0 \uc11c\ub85c \ub2e4\ub978 \uac12 \\(x_0,\\) \\(x_1\\)\uc744 \uc0dd\uac01\ud558\uace0<br \/>\n\\[y_0 = f(x_0 ) ,\\quad y_1 = f(x_1 )\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(x\\)\uac00 \\(x_0\\)\uc5d0\uc11c \\(x_1\\)\uae4c\uc9c0\uc758 \uac12\uc744 \ucde8\ud55c\ub2e4\uba74 <span class=\"defined\">\\(\\boldsymbol x\\)\uc758 \uc99d\uac00\ub7c9<\/span>\uc740<br \/>\n\\[\\Delta x = x_1 &#8211; x_0\\]<br \/>\n\uc774\uace0, <span class=\"defined\">\\(\\boldsymbol y\\)\uc758 \uc99d\uac00\ub7c9<\/span>\uc740<br \/>\n\\[\\Delta y = y_1 &#8211; y_0 = f(x_1 ) &#8211; f(x_0 ) = f(x_0 + \\Delta x) &#8211; f(x_0 )\\]<br \/>\n\uc774\ub2e4. \uc774\ub54c, \\(x\\)\uc758 \uac12\uc774 \\(x_0\\)\uc5d0\uc11c \uc2dc\uc791\ud558\uc5ec \\(x_1\\)\uc5d0 \uc774\ub974\ub294 \ub3d9\uc548 \ud568\uc218 \\(f\\)\uc758 <span class=\"defined\">\ud3c9\uade0\ubcc0\ud654\uc728<\/span>(average rate of change)\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\frac{\\Delta y}{\\Delta x} = \\frac{f(x_1 ) &#8211; f(x_0 )}{x_1 &#8211; x_0} = \\frac{f(x_0 + \\Delta x ) -f(x_0 )}{\\Delta x} .\\]<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.1.1.<\/span><br \/>\n\ud568\uc218 \\(f(x) = x^2 -2x+5\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(x\\)\uc758 \uac12\uc774 \\(3\\)\uc5d0\uc11c \uc2dc\uc791\ud558\uc5ec \\(3.2\\)\uc5d0 \uc774\ub974\ub294 \ub3d9\uc548 \\(f\\)\uc758 \ud3c9\uade0\ubcc0\ud654\uc728\uc744 \uad6c\ud574 \ubcf4\uc790. \\(x\\)\uc758 \uac12\uc758 \uc99d\uac00\ub7c9\uacfc \\(y\\)\uc758 \uac12\uc758 \uc99d\uac00\ub7c9\uc740 \uac01\uac01 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\Delta x &#038;= 0.2 , \\\\[6pt]<br \/>\n\\Delta y &#038;= f(3.2) &#8211; f(3) = 8.84 &#8211; 8 = 0.84.<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uc758 \ud3c9\uade0\ubcc0\ud654\uc728\uc740<br \/>\n\\[\\frac{\\Delta y}{\\Delta x} = \\frac{0.84}{0.2} = 4.2\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\ud568\uc218 \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504 \uc704\uc758 \uc810 \\((x_0 ,\\, f(x_0 ))\\)\uc5d0\uc11c \uc774 \uadf8\ub798\ud504\uc5d0 \uc811\ud558\ub294 \uc9c1\uc120\uc758 \uae30\uc6b8\uae30\ub97c \uad6c\ud574 \ubcf4\uc790. \uc544\ub798 \uadf8\ub9bc\uc744 \uc0b4\ud3b4 \ubcf4\uc790.<\/p>\n<div><a href=\"\/blog\/wp-content\/uploads\/2021\/09\/ic_501_1.png\" data-rel=\"penci-gallery-image-content\" ><img decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2021\/09\/ic_501_1.png\" alt=\"\" width=\"248\" height=\"187\" class=\"aligncenter size-full wp-image-7790\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_501_1.png 744w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_501_1-300x227.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_501_1-585x442.png 585w\" sizes=\"(max-width: 248px) 100vw, 248px\" \/><\/a><\/div>\n<p>\ud568\uc218\uc758 \uadf8\ub798\ud504\uac00 \ub9e4\ub044\ub7fd\ub2e4\uba74 \uc811\uc120\uc758 \uae30\uc6b8\uae30\ub294 \ub2e4\uc74c \uadf9\ud55c\uac12\uacfc \uac19\ub2e4.<br \/>\n\\[\\lim_{\\Delta x \\rightarrow 0} \\frac{f(x_0 + \\Delta x) &#8211; f(x_0 )}{\\Delta x}\\]<br \/>\n\ub9cc\uc57d \uc774 \uadf9\ud55c\uc774 \uc218\ub834\ud55c\ub2e4\uba74, \uadf8 \uadf9\ud55c\uac12\uc744 \\(x_0\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\ubbf8\ubd84\uacc4\uc218<\/span>(derivative)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\boldsymbol{f &#8216; (x_0 )}\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ubbf8\ubd84\uacc4\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uae30\ud638\ub294 \ub2e4\uc591\ud558\ub2e4. \ub2e4\uc74c \uae30\ud638\ub294 \ubaa8\ub450 \\(x_0\\)\uc5d0\uc11c \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uae30\ud638\uc774\ub2e4.<br \/>\n\\[\\begin{gather}<br \/>\nf &#8216; (x_0 ) \\quad\\quad<br \/>\n\\frac{d}{dx} f(x_0 ) \\quad\\quad<br \/>\n\\frac{df}{dx} (x_0 ) \\quad\\quad<br \/>\n\\left. \\frac{df}{dx} \\right\\vert _{x=x_0}  \\\\[7pt]<br \/>\n\\left. \\frac{df}{dx} \\right\\vert _{x-x_0} \\quad\\quad<br \/>\nDf(x_0 ) \\quad\\quad<br \/>\nDf \\bigg\\vert _{x=x_0 }<br \/>\n\\end{gather}\\]<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 5.1.2.<\/span><br \/>\n\\(x_0 = 3\\)\uc5d0\uc11c \ud568\uc218 \\(f(x) = 2x^2 &#8211; 3x+5\\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n<p class=\"margintop1\"><span class=\"proof\">\ud480\uc774.<\/span><br \/>\n\\(\\Delta x \\ne 0\\)\uc77c \ub54c<br \/>\n\\[\\begin{align}<br \/>\nf(3+\\Delta x ) &#8211; f(3) &#038;= 2 (\\Delta x)^2 + 9 \\Delta x , \\\\[6pt]<br \/>\n\\frac{f(3+\\Delta x) &#8211; f(3)}{\\Delta x} &#038;= \\frac{2(\\Delta x)^2 + 9 \\Delta x}{\\Delta x} = 2\\Delta x+9<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\lim_{\\Delta x \\rightarrow 0} \\frac{f(3+\\Delta x) -f(3)}{\\Delta x} = \\lim_{\\Delta x \\rightarrow 0} (2\\Delta x + 9 ) = 9\\]<br \/>\n\uc989 \\(f &#8216; (3) = 9\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 5.1.3.<\/span> \\(x_0 = 3\\)\uc77c \ub54c \ud568\uc218 \\(y=2x^2 -3x+5\\)\uc758 \uadf8\ub798\ud504\uc5d0 \uc811\ud558\ub294 \uc9c1\uc120\uc758 \ubc29\uc815\uc2dd\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n<p class=\"margintop1\"><span class=\"proof\">\ud480\uc774.<\/span><br \/>\n\\(x=3\\)\uc77c \ub54c \\(y=14\\)\uc774\ubbc0\ub85c \uc811\uc810\uc758 \uc88c\ud45c\ub294 \\((3,\\,14)\\)\uc774\ub2e4.<br \/>\n\\[\\left. \\frac{dy}{dx} \\right\\vert_{x=3} = 9\\]<br \/>\n\uc774\ubbc0\ub85c \uc811\uc120\uc758 \uae30\uc6b8\uae30\ub294 \\(9\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc740<br \/>\n\\[y = 9(x-3)+14\\]<br \/>\n\uc774\ub2e4. \uc774 \uc2dd\uc744 \uac04\ub2e8\ud558\uac8c \ub098\ud0c0\ub0b4\uba74<br \/>\n\\[y=9x-13\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc810 \\(x_0\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218\uac00 \uadf9\ud55c\uc73c\ub85c \uc815\uc758\ub418\ubbc0\ub85c, \\(x_0\\)\ub294 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc5d0 \uc18d\ud558\uba74\uc11c \ub3d9\uc2dc\uc5d0 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc9d1\uc801\uc810\uc774\uc5b4\uc57c \ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.1.4.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\ud568\uc218 \\(f\\)\uac00 \\(\\mathbb{Z}\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uba74 \\(\\mathbb{Z}\\)\uc758 \uc5b4\ub290 \uc810\uc5d0\uc11c\ub3c4 \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218\uac00 \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(\\mathbb{Z}\\)\ub294 \uc9d1\uc801\uc810\uc744 \uac16\uc9c0 \uc54a\uae30 \ub54c\ubb38\uc774\ub2e4.<\/li>\n<li>\\((a,\\,b)\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uc5f4\ub9b0\uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\((a,\\,b)\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uba74 \\((a,\\,b)\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218\uac00 \uc815\uc758\ub41c\ub2e4. (\uc815\uc758\ub41c\ub2e4\uace0 \ud574\uc11c \ubc18\ub4dc\uc2dc \uc874\uc7ac\ud558\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4.) \uc5f4\ub9b0\uad6c\uac04\uc758 \ub05d\uc810 \\(a\\)\uc640 \\(b\\)\uc5d0\uc11c\ub294 \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218\uac00 \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<li>\\([a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \ub2eb\ud78c\uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5c8\ub2e4\uba74 \\([a,\\,b]\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218\uac00 \uc815\uc758\ub41c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\ub9cc\uc57d \\(f\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \ub2eb\ud78c\uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uba74, \ub2e4\uc74c \ub450 \uadf9\ud55c\uc740 \uac19\uc740 \uadf9\ud55c\uc744 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[\\lim_{\\Delta x \\rightarrow 0} \\frac{f(a+\\Delta x ) &#8211; f(a)}{\\Delta x} ,\\quad<br \/>\n\\lim_{\\Delta x \\rightarrow 0^+} \\frac{f(a+\\Delta x ) &#8211; f(a)}{\\Delta x}.\\]<br \/>\n\ub9cc\uc57d \ub450 \ubc88\uc9f8 \uadf9\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4\uba74, \uc774 \uadf9\ud55c\uc744 \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc6b0\ubbf8\ubd84\uacc4\uc218<\/span>(right-hand derivative)\ub77c\uace0 \ubd80\ub974\uace0 \\(f_r &#8216; (a)\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \\(b\\)\uc5d0\uc11c \\(f\\)\uc758 \uc88c\ubbf8\ubd84\uacc4\uc218\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ub41c\ub2e4. \uc989 \ub9cc\uc57d \uadf9\ud55c<br \/>\n\\[\\lim_{\\Delta x \\rightarrow 0^-} \\frac{f(b + \\Delta x) -f(b)}{\\Delta x}\\]<br \/>\n\uac00 \uc874\uc7ac\ud55c\ub2e4\uba74, \uc774 \uadf9\ud55c\uc744 \\(b\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc88c\ubbf8\ubd84\uacc4\uc218<\/span>(left-hand derivative)\ub77c\uace0 \ubd80\ub974\uace0 \\(f_l &#8216; (b)\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ub9cc\uc57d \uc810 \\(x_0\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218\uac00 \uc874\uc7ac\ud558\uba74 \u201c\\(f\\)\uac00 \\(x_0\\)\uc5d0\uc11c <span class=\"defined\">\ubbf8\ubd84 \uac00\ub2a5<\/span>\ud558\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(E\\)\uac00 \ud568\uc218 \\(f:D \\rightarrow \\mathbb{R}\\)\uc758 \uc815\uc758\uc5ed\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(f\\)\uac00 \\(E\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \u201c\\(f\\)\uac00 \\(E\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\u201d\ub77c\uace0 \ub9d0\ud55c\ub2e4. \uc774 \uacbd\uc6b0 \\(f &#8216; \\)\uc740 \\(E\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\uac00 \ub418\ub294\ub370, \uc774 \ud568\uc218 \\(f &#8216; \\)\uc744 \\(f\\)\uc758 <span class=\"defined\">\ub3c4\ud568\uc218<\/span>(derivative, derived function)\ub77c\uace0 \ubd80\ub978\ub2e4. \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\ub294 \uac83\uc744 <span class=\"defined\">\ubbf8\ubd84<\/span>(differentiation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \u201c\ub3c4\ud568\uc218\ub97c \uad6c\ud55c\ub2e4\u201d\ub77c\ub294 \ub9d0\uacfc \u201c\ubbf8\ubd84\ud55c\ub2e4\u201d\ub77c\ub294 \ub9d0\uc740 \uac19\uc740 \ub73b\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\uc608\uc81c 5.1.5.<\/span><br \/>\n\ub2e4\uc74c \ud568\uc218\ub97c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud558\uc2dc\uc624.<br \/>\n(1) \\(f(x) = 3x+1\\) <br \/>\n(2) \\(f(x) = x^2\\) <br \/>\n(3) \\(f(x) = \\sqrt{x}\\) <br \/>\n(4) \\(f(t) = \\sin t\\) <\/p>\n<p class=\"margintop1\"><span class=\"proof\">\ud480\uc774.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\uc2e4\uc218 \\(x\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[\\begin{align}<br \/>\nf &#8216; (x) &#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x) &#8211; f(x)}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{(3x + 3\\Delta +1) &#8211; (3x+1)}{\\Delta x} = 3.<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uac00 \\(\\mathbb{R}\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(f &#8216; (x) = 3\\)\uc774\ub2e4.<\/li>\n<li style=\"margin-top: 0.7em;\">\uc2e4\uc218 \\(x\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[\\begin{align}<br \/>\nf &#8216; (x) &#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+ \\Delta x) &#8211; f(x)}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{(x+\\Delta x)^2 &#8211; x^2}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} (2x+\\Delta x) = 2.<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uac00 \\(\\mathbb{R}\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(f &#8216; (x) = 2x\\)\uc774\ub2e4.<\/li>\n<li style=\"margin-top: 0.7em;\">\ud568\uc218 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc740 \\([0,\\,\\infty )\\)\uc774\ub2e4. \ub9cc\uc57d \\(x > 0\\)\uc774\uba74<br \/>\n\\[\\begin{align}<br \/>\nf &#8216; (x) &#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x) &#8211; f(x)}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{\\sqrt{x+\\Delta x} &#8211; \\sqrt{x}}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{(\\sqrt{x+\\Delta x} &#8211; \\sqrt{x})(\\sqrt{x+\\Delta x} + \\sqrt{x})}{\\Delta x (\\sqrt{x+\\Delta x} + \\sqrt{x})} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{1}{\\sqrt{x+\\Delta x} + \\sqrt{x}} = \\frac{1}{2\\sqrt{x}}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \ub9cc\uc57d \\(x=0\\)\uc774\uba74<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x ) &#8211; f(x)}{\\Delta x} = \\lim_{\\Delta x \\rightarrow 0} \\frac{\\sqrt{\\Delta x}}{\\Delta x} = \\lim_{\\Delta x \\rightarrow 0} \\frac{1}{\\sqrt{\\Delta x}} = \\infty<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uc989 \\(f\\)\ub294 \\(0\\)\uc5d0\uc11c \ubbf8\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4.<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(f\\)\ub294 \\((0,\\,\\infty)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \ub3c4\ud568\uc218\ub294<br \/>\n\\[f &#8216; (x) = \\frac{1}{2\\sqrt{x}}\\]<br \/>\n\uc774\ub2e4. \\(f\\)\uc758 \ub3c4\ud568\uc218\uc758 \uc815\uc758\uc5ed\uc774 \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc9c4\ubd80\ubd84\uc9d1\ud569\uc774\ub77c\ub294 \uc0ac\uc2e4\uc774 \ud765\ubbf8\ub86d\ub2e4.<\/li>\n<li style=\"margin-top: 0.7em;\">\\(f(t)\\)\uc5d0 \ubcc0\uc218 \\(x\\)\uac00 \uc5c6\ub2e4. \uc989 \\(x\\)\ub97c \ubcc0\uc218\ub85c \ubcf4\uc558\uc744 \ub54c \\(f(t)\\)\ub294 \uc0c1\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\frac{d}{dx} f(t) =0\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\ud568\uc218 \\(f : D \\rightarrow \\mathbb{R}\\)\uac00 \uc810 \\(x_0 \\in D\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\begin{align}<br \/>\n\\lim_{x\\rightarrow x_0} f(x)<br \/>\n&#038;= \\lim_{x\\rightarrow x_0} ( f(x) &#8211; f(x_0 ) + f(x_0 ) ) \\\\[4pt]<br \/>\n&#038;= \\lim_{x\\rightarrow x_0} \\left\\{ \\frac{f(x)-f(x_0 )}{x-x_0} (x-x_0 ) \\right\\} + f(x_0 ) \\\\[4pt]<br \/>\n&#038;= \\lim_{x\\rightarrow x_0} \\left\\{ \\frac{f(x)-f(x_0 )}{x-x_0} \\right\\} \\times \\lim_{x\\rightarrow x_0} (x-x_0 ) + f(x_0 ) \\\\[4pt]<br \/>\n&#038;= f &#8216; (x_0 ) \\times 0 + f(x_0) \\\\[5pt]<br \/>\n&#038;= f(x_0 )<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uc989 \\(x_0\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf9\ud55c\uac12\uacfc \ud568\uc22b\uac12\uc774 \uc77c\uce58\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uac00 \\(x_0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 5.1.1. (\uc5f0\uc18d\uc131\uacfc \ubbf8\ubd84 \uac00\ub2a5\uc131\uc758 \uad00\uacc4)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uc810 \\(x_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(f\\)\uac00 \\(x_0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"remark\">\n<p><span class=\"remark\">\ucc38\uace0.<\/span><br \/>\n\uc5f0\uc18d\ud568\uc218\uac00 \ubaa8\ub450 \ubbf8\ubd84 \uac00\ub2a5\ud55c \uac83\uc740 \uc544\ub2c8\ub2e4. \uc608\ucee8\ub300 \ud568\uc218 \\(f:\\mathbb{R} \\rightarrow \\mathbb{R}\\)\uac00 \\[f(x) = \\lvert x \\rvert\\]\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uba74 \\(f\\)\ub294 \\(0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uc9c0\ub9cc \\(0\\)\uc5d0\uc11c \ubbf8\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4.<br \/>\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<div class=\"remark\">\n<p><span class=\"remark\">\ucc38\uace0.<\/span><br \/>\n\\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(f\\)\uac00 \uc5f0\uc18d\uc774\uc9c0\ub9cc \\(f &#8216; \\)\uc740 \uc5f0\uc18d\uc774 \uc544\ub2d0 \uc218 \uc788\ub2e4. \uc608\ucee8\ub300 \ud568\uc218 \\(f:\\mathbb{R} \\rightarrow \\mathbb{R}\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[f(x) = \\begin{cases}<br \/>\nx^2 \\sin \\frac{1}{x} &#038; \\quad \\text{if} \\quad x\\ne 0 , \\\\[4pt]<br \/>\n0 &#038; \\quad \\text{if} \\quad x=0.<br \/>\n\\end{cases}\\]<br \/>\n\uc774 \ud568\uc218\ub294 \\(\\mathbb{R}\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \\(f &#8216; \\)\uc740 \\(0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2c8\ub2e4. \uc9c0\uae08\uae4c\uc9c0 \ub2e4\ub8ec \ub0b4\uc6a9\ub9cc\uc73c\ub85c\ub294 \uc774 \uc0ac\uc2e4\uc744 \uc99d\uba85\ud560 \uc218 \uc5c6\uc9c0\ub9cc 5\uc7a5 4\uc808\uae4c\uc9c0 \ub9c8\uce5c \ud6c4\uc5d0\ub294 \uc774 \uc0ac\uc2e4\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4. \ub3c4\uc804!<br \/>\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<p>\uc774\uc81c \ub3c4\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uc0b4\ud3b4\ubcf4\uc790. \ud568\uc218 \\(f\\)\uac00 \\(x_0\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0\uad6c\uac04\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f &#8216; \\)\uc774 \\(x_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(x_0\\)\uc5d0\uc11c \\(f &#8216; \\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[f &#8216; &#8216; (x_0 ) = \\left. \\frac{df &#8216; }{dx} \\right\\vert_{x=x_0} = \\lim_{\\Delta x \\rightarrow 0} \\frac{f &#8216; (x_0 + \\Delta x) &#8211; f &#8216; (x_0 )}{\\Delta x}.\\]<br \/>\n\uc774 \uac12\uc744 \\(x_0\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc774\uacc4\ubbf8\ubd84\uacc4\uc218<\/span>(second derivative)\ub77c\uace0 \ubd80\ub978\ub2e4. \\(x_0\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uacc4\ubbf8\ubd84\uacc4\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uae30\ud638\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ub2e4\uc591\ud558\ub2e4.<br \/>\n\\[\\begin{gather}<br \/>\nf^{(2)}(x_0) \\quad\\quad<br \/>\n\\frac{d^2}{(dx)^2} f(x_0 ) \\quad\\quad<br \/>\n\\frac{d^2 f}{(dx)^2} (x_0 ) \\quad\\quad<br \/>\n\\left. \\frac{d^2 f}{(dx)^2} \\right\\vert_{x=x_0} \\\\[6pt]<br \/>\n\\left. \\frac{d^2 y}{(dx)^2} \\right\\vert_{x=x_0} \\quad\\quad<br \/>\nD^2 f(x_0 ) \\quad\\quad<br \/>\nD^2 f \\bigg\\vert_{x=x_0}<br \/>\n\\end{gather}\\]<br \/>\n\uac19\uc740 \ubc29\ubc95\uc73c\ub85c \\(n\\)\uc774 \\(2\\) \uc774\uc0c1\uc778 \uc790\uc5f0\uc218\uc77c \ub54c \\(x_0\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\\(\\boldsymbol{n}\\)\uacc4\ubbf8\ubd84\uacc4\uc218<\/span>\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \\(x_0\\)\uc5d0\uc11c \\(f\\)\uc758 \\(n\\)\uacc4\ubbf8\ubd84\uacc4\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uae30\ud638\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ub2e4\uc591\ud558\ub2e4.<br \/>\n\\[\\begin{gather}<br \/>\nf^{(n)} (x_0 ) \\quad\\quad<br \/>\n\\frac{d^n}{(dx)^n} f(x_0 ) \\quad\\quad<br \/>\n\\frac{d^n f}{(dx)^n} (x_0 ) \\quad\\quad<br \/>\n\\left. \\frac{d^n f}{(dx)^n} \\right\\vert_{x=x_0} \\\\[6pt]<br \/>\n\\left. \\frac{d^n y}{(dx)^n} \\right\\vert_{x=x_0} \\quad\\quad<br \/>\nD^n f(x_0 ) \\quad\\quad<br \/>\nD^n f \\bigg\\vert_{x=x_0}<br \/>\n\\end{gather}\\]\n<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.1.6.<\/span><br \/>\n\\(f(x)=x^3\\)\uc77c \ub54c \\(f\\)\uc758 \\(n\\)\uacc4\ubbf8\ubd84\uacc4\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nf &#8216; (x) &#038;= 3x^2 ,\\\\[5pt]<br \/>\nf &#8216; &#8216; (x) &#038;= 6x , \\\\[5pt]<br \/>\nf ^{(3)} (x) &#038;= 6 , \\\\[5pt]<br \/>\nf^{(4)} (x) &#038;= 0 , \\\\[5pt]<br \/>\nf^{(5)} (x) &#038;= 0 , \\\\[5pt]<br \/>\n\\quad\\quad\\vdots \\\\[5pt]<br \/>\nf^{(n)} (x) &#038; =0 \\quad \\text{for} \\,\\, n \\ge 4.<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p>\\(I\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\ub77c\uace0 \ud558\uc790. (\uae38\uc774\uac00 \ubb34\ud55c\ub300\uc778 \uacbd\uc6b0\ub97c \ud3ec\ud568\ud558\uc790.) \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(n\\)\uacc4\ubbf8\ubd84\uacc4\uc218\uac00 \uc874\uc7ac\ud558\ub294 \uc2e4\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc744 \\(D^{(n)} (I)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub610\ud55c \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \uc5f0\uc18d\uc778 \\(n\\)\uacc4\ub3c4\ud568\uc218\ub97c \uac16\ub294 \uc2e4\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc744 \\(C^{(n)} (I)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\(n=1\\)\uc774\uba74 \\(D^{(1)} (I)\\)\ub97c \\(D(I)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \\(D(I)\\)\ub294 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c \uc2e4\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc774\ub2e4. \ub9cc\uc57d \\(n=0\\)\uc774\uba74 \\(C^{(0)} (I)\\)\ub97c \\(C(I)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989 \\(C(I)\\)\ub294 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \uc2e4\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc774\ub2e4.<\/p>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n\n\n\n<p>.<\/p>\n\n\n\n\n\n<p>.<\/p>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/vector-valued-functions\">\ubca1\ud130\ud568\uc218\uc758 \uadf9\ud55c<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/differentiation-theorems\">\ubbf8\ubd84 \uacf5\uc2dd<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 5\uc7a5 1\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \ud568\uc218 \\(y=f(x)\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \\(x\\)\uc640 \\(y\\)\ub294 \uc2e4\uc218 \ubcc0\uc218\uc774\ub2e4. \uadf8\ub9ac\uace0 \uc11c\ub85c \ub2e4\ub978 \uac12 \\(x_0,\\) \\(x_1\\)\uc744 \uc0dd\uac01\ud558\uace0 \\(y_0 = f(x_0 ) ,\\, y_1 = f(x_1 )\\) \uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(x\\)\uac00 \\(x_0\\)\uc5d0\uc11c \\(x_1\\)\uae4c\uc9c0\uc758 \uac12\uc744 \ucde8\ud55c\ub2e4\uba74 \\(\\boldsymbol x\\)\uc758 \uc99d\uac00\ub7c9\uc740 \\(\\Delta x = x_1 &#8211; x_0\\) \uc774\uace0, \\(\\boldsymbol y\\)\uc758 \uc99d\uac00\ub7c9\uc740 \\(\\Delta y = y_1 &#8211; y_0 =&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":501,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6712","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6712","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=6712"}],"version-history":[{"count":30,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6712\/revisions"}],"predecessor-version":[{"id":7996,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6712\/revisions\/7996"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6712"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}