{"id":6718,"date":"2021-07-21T00:04:31","date_gmt":"2021-07-20T15:04:31","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6718"},"modified":"2022-03-06T19:35:57","modified_gmt":"2022-03-06T10:35:57","slug":"elementary-transcendental-functions","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/elementary-transcendental-functions\/","title":{"rendered":"\uae30\ubcf8 \ucd08\uc6d4\ud568\uc218\uc758 \ubbf8\ubd84"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 5\uc7a5 4\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

\uc0ac\uce59\uacc4\uc0b0\uacfc \uc81c\uacf1\uadfc\uc744 \uc720\ud55c \ubc88 \uc0ac\uc6a9\ud558\uc5ec \ub098\ud0c0\ub0bc \uc218 \uc788\ub294 \ud568\uc218\ub97c \ub300\uc218\uc801 \ud568\uc218\ub77c\uace0 \ubd80\ub974\uba70, \ub300\uc218\uc801 \ud568\uc218\uac00 \uc544\ub2cc \ud568\uc218\ub97c \ucd08\uc6d4\ud568\uc218\ub77c\uace0 \ubd80\ub978\ub2e4. \uc0bc\uac01\ud568\uc218, \uc9c0\uc218\ud568\uc218, \ub85c\uadf8\ud568\uc218\ub294 \ub300\ud45c\uc801\uc778 \ucd08\uc6d4\ud568\uc218\uc774\ub2e4. \ubcf4\ud1b5 \u2018\ucd08\ub4f1\ud568\uc218\u2019 \ub610\ub294 \u2018\uae30\ubcf8 \ucd08\uc6d4\ud568\uc218\u2019\ub77c\uace0 \ud558\uba74 \uc774\ub4e4 \uc138 \uc885\ub958\uc758 \ucd08\uc6d4\ud568\uc218\ub97c \uc774\ub978\ub2e4.<\/p>\n

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc0bc\uac01\ud568\uc218, \uc9c0\uc218\ud568\uc218, \ub85c\uadf8\ud568\uc218\uc758 \ubbf8\ubd84\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

<\/p>\n

\uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84<\/h2>\n

\uc608\uc81c 3.2.6\uacfc \uc608\uc81c 3.2.7\uc5d0\uc11c \ub2e4\uc74c \ub450 \uacf5\uc2dd\uc744 \uc720\ub3c4\ud558\uc600\ub2e4.
\n\\[\\lim_{x\\rightarrow 0}\\frac{\\sin x}{x} = 1,\\quad \\lim_{x\\rightarrow 0} \\frac{1-\\cos x}{x} =0.\\]
\n\uc774 \uadf9\ud55c\uc744 \uc774\uc6a9\ud558\uc5ec \uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc744 \uc720\ub3c4\ud574 \ubcf4\uc790.<\/p>\n

\uba3c\uc800 \uc0ac\uc778\uc758 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\begin{align}
\n\\frac{d}{dx}\\sin x
\n&= \\lim_{\\Delta x \\rightarrow 0} \\frac{\\sin (x+\\Delta x) – \\sin x}{\\Delta x} \\\\[4pt]
\n&= \\lim_{\\Delta x \\rightarrow 0} \\frac{\\sin x \\cos \\Delta x + \\cos x \\sin \\Delta x – \\sin x}{\\Delta x} \\\\[4pt]
\n&= \\lim_{\\Delta x \\rightarrow 0} \\frac{\\cos x \\sin \\Delta x – \\sin x ( 1- \\cos \\Delta x)}{\\Delta x} \\\\[4pt]
\n&= \\lim_{\\Delta x \\rightarrow 0} \\left\\{ \\cos x \\cdot \\frac{\\sin \\Delta x}{\\Delta x}\\right\\} – \\lim_{\\Delta x \\rightarrow 0} \\left\\{ \\sin x \\cdot \\frac{1-\\cos \\Delta x}{\\Delta x}\\right\\} \\\\[6pt]
\n&= \\cos x \\cdot 1 – \\sin x \\cdot 0 \\\\[6pt]
\n&= \\cos x.
\n\\end{align}\\]
\n\ucf54\uc0ac\uc778\uacfc \ud0c4\uc820\ud2b8\uc758 \ub3c4\ud568\uc218\ub294 \uc0ac\uc778\uc758 \ub3c4\ud568\uc218\ub85c\ubd80\ud130 \uc720\ub3c4\ub41c\ub2e4.
\n\\[\\begin{align}
\n\\frac{d}{dx} \\cos x
\n&= \\frac{d}{dx} \\sin \\left( x+ \\frac{\\pi}{2}\\right) \\\\[4pt]
\n&= \\cos \\left( x+\\frac{\\pi}{2}\\right) \\\\[6pt]
\n&= – \\sin x , \\\\[8pt]
\n\\frac{d}{dx} \\tan x
\n&= \\frac{d}{dx} \\left\\{ \\frac{\\sin x}{\\cos x}\\right\\} \\\\[4pt]
\n&= \\frac{\\cos x \\cdot \\cos x – \\sin x \\cdot (-\\sin x)}{(\\cos x)^2} \\\\[4pt]
\n&= \\frac{1}{(\\cos x)^2} \\\\[6pt]
\n&= \\sec^2 x .
\n\\end{align}\\]
\n\ub2e4\ub978 \uc0bc\uac01\ud568\uc218(\uc2dc\ucee8\ud2b8, \ucf54\uc2dc\ucee8\ud2b8, \ucf54\ud0c4\uc820\ud2b8)\uc758 \ub3c4\ud568\uc218\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc720\ub3c4\ub41c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 5.4.1. (\uc0bc\uac01\ud568\uc218\uc758 \ub3c4\ud568\uc218)<\/span><\/p>\n

\\[\\begin{array}{lll}
\n\\displaystyle\\frac{d}{dx} \\sin x = \\cos x & \\quad\\quad & \\displaystyle\\frac{d}{dx}\\sec x = \\tan x \\sec x \\\\[6pt]
\n\\displaystyle\\frac{d}{dx} \\cos x = – \\sin x & \\quad\\quad & \\displaystyle\\frac{d}{dx} \\csc x = – \\csc x \\cot x \\\\[6pt]
\n\\displaystyle\\frac{d}{dx} \\tan x = \\sec^2 x & \\quad\\quad & \\displaystyle\\frac{d}{dx} \\cot x = -\\csc^2 x
\n\\end{array}\\]\n<\/p>\n<\/div>\n

\n

\ubcf4\uae30 5.4.1.<\/span><\/p>\n

    \n
  1. \\(y = \\sin x + \\cos x\\)\uc774\uba74
    \n\\[y ‘ = \\cos x – \\sin x\\]
    \n\uc774\ub2e4.<\/li>\n
  2. \\(y = \\sin x \\cos x\\)\uc774\uba74
    \n\\[y ‘ = (\\sin x) ‘ \\cos x + \\sin x (cos x )’ = \\cos^2 x – \\sin^2 x = \\cos 2x\\]
    \n\uc774\ub2e4.<\/li>\n
  3. \\(y = \\displaystyle\\frac{1}{\\cos x}\\)\uc774\uba74
    \n\\[y ‘ = \\frac{\\sin x}{\\cos^2 x} = \\frac{\\sin x}{\\cos x} \\cdot \\frac{1}{\\cos x} = \\tan x \\sec x\\]
    \n\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

    <\/p>\n

    \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \uadf9\ud55c<\/h2>\n

    \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uc720\ub3c4\ud558\uae30 \uc704\ud574 \ud544\uc694\ud55c \uadf9\ud55c\uc744 \uc0b4\ud3b4 \ubcf4\uc790. 2\uc7a5 2\uc808(\uc591\ud56d\uae09\uc218)\uc5d0\uc11c \uc790\uc5f0\uc0c1\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc600\ub2e4.
    \n\\[\\lim_{n\\rightarrow\\infty} \\left( 1+\\frac{1}{n}\\right)^n = e.\\]
    \n\uc5ec\uae30\uc11c \uc790\uc5f0\uc218 \\(n\\)\uc744 \uc2e4\uc218 \\(x\\)\ub85c \ubc14\uafbc \ud568\uc218\uc758 \uadf9\ud55c\ub3c4 \uac19\uc740 \uac12\uc5d0 \uc218\ub834\ud568\uc774 \uc54c\ub824\uc838 \uc788\ub2e4. \uc989
    \n\\[\\lim_{x\\rightarrow\\infty} \\left(1+\\frac{1}{x}\\right)^x = e\\]
    \n\uc774\ub2e4. \ub9cc\uc57d \uc704 \uadf9\ud55c\uc5d0\uc11c \\(x\\)\ub97c \\(1\/t\\)\ub85c \ubc14\uafb8\uba74 \\(x\\rightarrow\\infty\\)\uc778 \uadf9\ud55c\uc774 \\(t\\rightarrow 0^+\\)\uc778 \uc6b0\uadf9\ud55c\uc73c\ub85c \ubc14\ub00c\uc5b4 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
    \n\\[\\lim_{t\\rightarrow 0^+} (1+t)^{\\frac{1}{t}} = e.\\]
    \n\ub9cc\uc57d \\(t < 0\\)\uc774\uace0 \\(s = -t\\)\uc774\uba74\n\\[\\begin{align}\n\\lim_{t\\rightarrow 0^-} (1+t)^{\\frac{1}{t}}\n&= \\lim_{s\\rightarrow 0^+} (1-s)^{-\\frac{1}{s}} \\\\[4pt]\n&= \\lim_{s\\rightarrow 0^+} \\left( \\frac{1}{1-s} \\right)^{\\frac{1}{s}} \\\\[4pt]\n&= \\lim_{s\\rightarrow 0^+} \\left( 1+ \\frac{s}{1-s} \\right)^{\\frac{1}{s}} \\\\[4pt]\n&= \\lim_{s\\rightarrow 0^+} \\left\\{ \\left( 1+ \\frac{s}{1-s} \\right)^{\\frac{1-s}{s}} \\right\\} ^{\\frac{1}{1-s}} .\n\\end{align}\\]\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(s\\rightarrow 0^+\\)\uc77c \ub54c\n\\[\\frac{s}{1-s} \\rightarrow 0^+ ,\\quad \\frac{1}{1-s} \\rightarrow 1\\]\n\uc774\ubbc0\ub85c\n\\[\\lim_{t\\rightarrow 0^-} (1+t)^{\\frac{1}{t}}\n=\n\\lim_{s\\rightarrow 0^+} \\left\\{ \\left( 1+ \\frac{s}{1-s} \\right)^{\\frac{1-s}{s}} \\right\\} ^{\\frac{1}{1-s}} = e^1 =e\\]\n\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \uc790\uc5f0\uc0c1\uc218 \uacf5\uc2dd (\ud568\uc218\uc758 \uadf9\ud55c)<\/span>
    \n\\[ \\lim_{x\\rightarrow\\infty} \\left( 1+ \\frac{1}{x}\\right)^x = \\lim_{x\\rightarrow 0} (1+x)^{\\frac{1}{x}} =e. \\]\n<\/p>\n<\/div>\n

    \ubc11\uc774 \\(e\\)\uc778 \uc9c0\uc218\ud568\uc218 \\(y=e^x\\)\uc744 \uc790\uc5f0\uc9c0\uc218\ud568\uc218<\/span>(natural exponential function) \ub610\ub294 \uac04\ub2e8\ud788 \uc9c0\uc218\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \ubc11\uc774 \\(e\\)\uc778 \ub85c\uadf8\ud568\uc218\ub97c \uc790\uc5f0\ub85c\uadf8\ud568\uc218<\/span>(natural logarithmic function)\ub77c\uace0 \ubd80\ub974\uace0 \\(\\ln x\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc989
    \n\\[\\ln x = \\log_e x \\quad\\text{for} \\,\\, x > 0\\]
    \n\uc774\ub2e4.<\/p>\n

    \uc790\uc5f0\ub85c\uadf8\ud568\uc218\uc640 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\ub294 \uc11c\ub85c \uc5ed\ud568\uc218 \uad00\uacc4\uc774\ub2e4. \uc989 \\(x > 0\\)\uc77c \ub54c
    \n\\[y=\\ln x \\quad \\Longleftrightarrow \\quad x=e^y\\]
    \n\uc774\ub2e4.<\/p>\n

    \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \uc131\uc9c8\uc744 \ubc1d\ud790 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \uadf9\ud55c \uacf5\uc2dd\uc774 \uc788\ub2e4. \uadf8 \uc911 \uccab \ubc88\uc9f8\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
    \n\\[\\begin{align}
    \n\\lim_{x\\rightarrow 0} \\frac{\\ln (1+x)}{x}
    \n&= \\lim_{x\\rightarrow 0} \\frac{1}{x} \\ln (1+x) \\\\[4pt]
    \n&= \\lim_{x\\rightarrow 0} \\ln (1+x)^{\\frac{1}{x}} \\\\[4pt]
    \n&= \\ln e \\\\[6pt]
    \n&= 1.
    \n\\end{align}\\]
    \n\ub450 \ubc88\uc9f8 \uadf9\ud55c \uacf5\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
    \n\\[\\lim_{x\\rightarrow 0} \\frac{e^x – 1}{x} = 1.\\]
    \n\uc774 \ub4f1\uc2dd\uc744 \uc720\ub3c4\ud574 \ubcf4\uc790. \\(t = e^x -1\\)\uc774\ub77c\uace0 \ud558\uba74 \\(e^x = 1+t\\)\uc774\uace0 \\(x = \\ln (1+t)\\)\uc774\ub2e4. \ub354\uc6b1\uc774 \\(t\\rightarrow 0\\)\uc77c \ub54c \\(x\\rightarrow 0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
    \n\\[\\begin{align}
    \n\\lim_{x\\rightarrow 0}\\frac{e^x -1}{x}
    \n&= \\lim_{t\\rightarrow 0} \\frac{t}{\\ln(1+t)} \\\\[4pt]
    \n&= \\lim_{t\\rightarrow 0} \\frac{1}{\\, \\displaystyle\\frac{\\ln (1+t)}{t} \\,} \\\\[4pt]
    \n&= \\frac{1}{\\displaystyle\\lim_{t\\rightarrow 0} \\ln (1+t)^{\\frac{1}{t}} } \\\\[4pt]
    \n&= \\frac{1}{\\ln e} \\\\[6pt]
    \n&= 1
    \n\\end{align}\\]
    \n\uc774\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 5.4.3. (\uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \uae30\ubcf8\uadf9\ud55c \uacf5\uc2dd)<\/span><\/p>\n

      \n
    1. \\(\\displaystyle \\lim_{x\\rightarrow 0} \\frac{e^x -1}{x} = 1.\\)<\/li>\n
    2. \\(\\displaystyle \\lim_{x\\rightarrow 0} \\frac{\\ln (1+x)}{x} = 1.\\)<\/li>\n<\/ol>\n<\/div>\n

      <\/p>\n

      \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ubbf8\ubd84<\/h2>\n

      \\(a > 0,\\) \\(a\\ne 1\\)\uc774\ub77c\uace0 \ud558\uc790. \uc790\uc5f0\ub85c\uadf8\ud568\uc218\uc640 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uac00 \uc11c\ub85c \uc5ed\ud568\uc218 \uad00\uacc4\uc774\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
      \n\\[\\begin{align}
      \ny=e^{x\\ln a} \\quad
      \n&\\Longleftrightarrow \\quad \\ln y = x \\ln a \\\\[6pt]
      \n&\\Longleftrightarrow \\quad x = \\frac{\\ln y}{\\ln a} = \\log_a y \\\\[6pt]
      \n&\\Longleftrightarrow \\quad y=a^x .
      \n\\end{align}\\]
      \n\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c \uacf5\uc2dd\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

      \n\\[a^x = e^{x\\ln a} \\quad\\quad (a > 0)\\]\n<\/div>\n

      \uc774\uc81c \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc744 \uc720\ub3c4\ud558\uc790.<\/p>\n

      \uba3c\uc800 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
      \n\\[\\begin{align}
      \n\\frac{d}{dx} e^x
      \n&= \\lim_{\\Delta x \\rightarrow 0} \\frac{e^{x+\\Delta x} – e^x}{\\Delta x} \\\\[4pt]
      \n&= \\lim_{\\Delta x \\rightarrow 0} \\frac{e^x ( e^{\\Delta x} -1)}{\\Delta x} \\\\[4pt]
      \n&= e^x \\cdot \\lim_{\\Delta x \\rightarrow 0} \\frac{e^{\\Delta x} -1}{\\Delta x} \\\\[6pt]
      \n&= e^x \\cdot 1 \\\\[6pt]
      \n&= e^x .
      \n\\end{align}\\]
      \n\uadf8\ub7f0\ub370 \\(a > 0,\\) \\(a\\ne 1\\)\uc77c \ub54c \\(a^x = e^{x\\ln a}\\)\uc774\ubbc0\ub85c, \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc73c\ub85c\ubd80\ud130 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
      \n\\[\\frac{d}{dx} a^x = \\frac{d}{dx} e^{x\\ln a} = e^{x\\ln a} \\cdot \\ln a = a^x \\ln a .\\]
      \n\ub2e4\uc74c\uc73c\ub85c \uc790\uc5f0\ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
      \n\\[\\begin{align}
      \n\\frac{d}{dx}\\ln x
      \n&= \\lim_{\\Delta x \\rightarrow 0} \\frac{\\ln (x+\\Delta x) – \\ln x}{\\Delta x} \\\\[4pt]
      \n&= \\lim_{\\Delta x \\rightarrow 0} \\frac{1}{\\Delta x} \\ln \\left( 1+ \\frac{\\Delta x}{x}\\right) \\\\[4pt]
      \n&= \\lim_{\\Delta x \\rightarrow 0} \\ln \\left( 1+ \\frac{\\Delta x}{x} \\right)^{\\frac{1}{\\Delta x}} \\\\[4pt]
      \n&= \\lim_{\\Delta x \\rightarrow 0} \\ln \\left\\{ \\left( 1+ \\frac{\\Delta x}{x} \\right)^{\\frac{x}{\\Delta x}} \\right\\} ^{\\frac{1}{x}} \\\\[4pt]
      \n&= \\frac{1}{x} \\cdot \\lim_{\\Delta x \\rightarrow 0} \\ln \\left( 1+ \\frac{\\Delta x}{x} \\right) ^{\\frac{x}{\\Delta x}} \\\\[4pt]
      \n&= \\frac{1}{x} \\cdot \\ln e \\\\[4pt]
      \n&= \\frac{1}{x}.
      \n\\end{align}\\]
      \n\uadf8\ub7f0\ub370 \\(a > 0,\\) \\(a \\ne 1,\\) \\(x > 0\\)\uc77c \ub54c
      \n\\[\\log_a x = \\frac{\\ln x}{\\ln a}\\]
      \n\uc774\ubbc0\ub85c, \uc790\uc5f0\ub85c\uadf8\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc73c\ub85c\ubd80\ud130 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
      \n\\[\\frac{d}{dx} \\log_a x = \\frac{d}{dx}\\left\\{ \\frac{\\ln x}{\\ln a} \\right\\} = \\frac{1}{x\\ln a} .\\]\n<\/p>\n

      \n

      \uc815\ub9ac 5.4.4. (\uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218)<\/span><\/p>\n

        \n
      1. \\(\\displaystyle\\frac{d}{dx} e^x = e^x \\)<\/li>\n
      2. \\(\\displaystyle\\frac{d}{dx} a^x = a^x \\ln a \\quad ( a > 0 ) \\)<\/li>\n
      3. \\(\\displaystyle\\frac{d}{dx} \\ln x = \\frac{1}{x} \\quad (x > 0 ) \\)<\/li>\n
      4. \\(\\displaystyle\\frac{d}{dx} \\log_a x = \\frac{1}{x\\ln a} \\quad (a > 0 ,\\,\\, a \\ne 1 ,\\,\\, x > 0) \\)<\/li>\n<\/ol>\n<\/div>\n
        \n

        \ubcf4\uae30 5.4.2.<\/span><\/p>\n

          \n
        1. \\(y=e^{2x-1}\\)\uc774\uba74 \\(y ‘ = 2e^{2x-1}\\)\uc774\ub2e4.<\/li>\n
        2. \\(y=3^{x+4}\\)\uc774\uba74 \\(y ‘ = 3^{x+4} \\ln 3\\)\uc774\ub2e4.<\/li>\n
        3. \\(y=e^x (\\sin x – \\cos x)\\)\uc774\uba74
          \n\\[y ‘ = e^x (\\sin x – \\cos x ) + e^x (\\cos x + \\sin x) = 2e^x \\sin x\\]
          \n\uc774\ub2e4.<\/li>\n
        4. \\(y=\\ln (x-5)\\)\uc774\uba74
          \n\\[y ‘ = \\frac{1}{x-5} \\quad \\text{for} \\,\\, x > 5\\]
          \n\uc774\ub2e4.<\/li>\n
        5. \\(y=\\log (5x+3)\\)\uc774\uba74
          \n\\[y ‘ = \\frac{1}{(5x+3)\\ln 10} \\quad\\text{for}\\,\\, x > – \\frac{3}{5}\\]
          \n\uc774\ub2e4.<\/li>\n
        6. \\(y = \\log_3 (2x^2 +1)\\)\uc774\uba74
          \n\\[y ‘ = \\frac{4x}{(2x^2 +1) \\ln 3}\\]
          \n\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \ubcf4\uae30 5.4.3.<\/span>
          \n\uc774\uc81c \ube44\ub85c\uc18c \uac70\ub4ed\uc81c\uacf1 \ubc95\uce59(\uc815\ub9ac 5.2.1)\uc744 \uc9c0\uc218\uac00 \uc2e4\uc218\uc778 \uacbd\uc6b0\ub85c \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4.<\/p>\n

          \\(\\alpha\\)\uac00 \ubb34\ub9ac\uc218\uc778 \uc0c1\uc218\uc774\uace0 \\(x > 0\\)\uc774\uba70 \\(f(x) = x^\\alpha\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74\\[f(x) = x^{\\alpha} = e^{\\alpha \\ln x}\\]
          \n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
          \n\\[f ‘ (x) = \\frac{d}{dx} (e^{\\alpha \\ln x} ) = e^{\\alpha \\ln x} \\cdot \\frac{\\alpha}{x} = x^{\\alpha} \\cdot \\frac{\\alpha}{x} = \\alpha x^{\\alpha -1}\\]
          \n\uc774\ub2e4.\n<\/p>\n<\/div>\n

          \n

          \uc608\uc81c 5.4.4.<\/span>
          \n\ub2e4\uc74c \ud568\uc218\ub97c \ubbf8\ubd84\ud558\uc2dc\uc624.<\/p>\n

          \\((1)\\,\\, \\displaystyle y=-\\frac{1}{x^2}\\)<\/div>\n
          \\((2) \\,\\, y = \\sqrt[3]{x^4}\\)<\/div>\n
          \\((3) \\,\\, y=\\sqrt{x-1}\\)<\/div>\n
          \\((4) \\,\\, y=x^e\\)<\/div>\n

          \ud480\uc774.<\/span><\/p>\n

            \n
          1. \\(y = -x^{-2}\\)\uc774\ubbc0\ub85c, \\(x\\ne 0\\)\uc77c \ub54c
            \n\\[y ‘ = 4x^{-3} = \\frac{4}{x^3}\\]
            \n\uc774\ub2e4.<\/li>\n
          2. \\(x > 0\\)\uc77c \ub54c \\(y=x^{\\frac{4}{3}}\\)\uc774\ubbc0\ub85c
            \n\\[y ‘ = \\frac{4}{3} x^{\\frac{1}{3}} = \\frac{4}{3} \\sqrt[3]{x}\\]
            \n\uc774\ub2e4. \\(x < 0\\)\uc77c \ub54c\ub294 \\(y = \\sqrt[3]{x^4} = \\sqrt[3]{(-x)^4} = (-x)^{\\frac{4}{3}}\\)\uc774\ubbc0\ub85c\n\\[y ' = - \\frac{4}{3} (-x)^{\\frac{1}{3}} = \\frac{4}{3} x^{\\frac{1}{3}} = \\frac{4}{3} \\sqrt[3]{x}\\]\n\uc774\ub2e4. \\(x=0\\)\uc77c \ub54c\ub294\n\\[y ' = \\lim_{h\\rightarrow 0} \\frac{\\sqrt[3]{h^4} - 0}{h} = 0\\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec\n\\[y ' = \\frac{4}{3} \\sqrt[3]{x}\\]\n\uc774\ub2e4.<\/li>\n
          3. \\(x \\ge 1\\)\uc77c \ub54c \\(y = (x-1)^{\\frac{1}{2}}\\)\uc774\ubbc0\ub85c, \\(x > 1\\)\uc77c \ub54c
            \n\\[y ‘ = \\frac{1}{2} (x-1)^{-\\frac{1}{2}} = \\frac{1}{2\\sqrt{x-1}}\\]
            \n\uc774\ub2e4. \\(x=1\\)\uc77c \ub54c\ub294 \\(y ‘ \\)\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4(\ubbf8\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4).<\/li>\n
          4. \\(x \\ge 0\\)\uc77c \ub54c \\(y ‘ = ex^{e-1}\\)\uc774\ub2e4. \\(x < 0\\)\uc77c \ub54c\ub294 \\(x^e\\)\uac00 (\uc2e4\uc218 \ubc94\uc704\uc5d0\uc11c) \uc815\uc758\ub418\uc9c0 \uc54a\ub294\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

            <\/p>\n

            \ucc38\uace0<\/h2>\n

            \uc0bc\uac01\ud568\uc218, \uc9c0\uc218\ud568\uc218, \ub85c\uadf8\ud568\uc218\uc758 \ubbf8\ubd84\uacfc \uad00\ub828\ub41c \ub354 \uc790\uc138\ud55c \ub0b4\uc6a9\uc744 \ubcf4\uace0 \uc2f6\ub2e4\uba74 \ub2e4\uc74c \uae00\uc744 \ucc38\uace0\ud558\uae30 \ubc14\ub780\ub2e4.<\/p>\n