미분공식

[eqn: (e ^{x} )'`=`e ^{x}]

[eqn: {BOX{해}}] [eqn: (e ^{x} ) prime][eqn: =` lim _{h -> 0} {} {e ^{x+h} `-`e ^{x}} over {h}]

[eqn: =e ^{x} ` lim _{h -> 0} {} {e ^{h} `-`1} over {h}]

[eqn: =e ^{x} ` lim _{t -> 1} {} {t`-`1} over {ln`t} ````````````````````````(e ^{h} `=`t)]

[eqn: =e ^{x} ` lim _{t -> 1} {} {1} over {{1} over {t-1} `ln`t}]

[eqn: =e ^{x} ` lim _{t -> 1} {} {1} over {ln`{(t) ^{t-1} }}]

[eqn: =e ^{x} ` lim _{t -> 1} {} {1} over {ln`e}]

[eqn: =e ^{x}]

[eqn: (a ^{x} ) prime `=`a ^{x} `ln`a]

[eqn: {BOX{해}}] [eqn: (a ^{x} ) prime][eqn: =` lim _{h -> 0} {} {a ^{x+h} `-`a ^{x}} over {h}]

[eqn: =a ^{x} ` lim _{h -> 0} {} {a ^{h} `-`1} over {h}]

[eqn: =a ^{x} ` lim _{t -> 1} {} {t`-`1} over {log _{a} `t} ````````````````````````(a ^{h} `=`t)]

[eqn: =a ^{x} ` lim _{t -> 1} {} {1} over {{1} over {t-1} `log _{a} `t}]

[eqn: =a ^{x} ` lim _{t -> 1} {} {1} over {log _{a} `(t) ^{t-1}}]

[eqn: =a ^{x} ` lim _{t -> 1} {} {1} over {log _{a} `e}]

[eqn: =a ^{x} `ln`a]

[eqn: (ln`x) prime `=` {1} over {x}]

[eqn: {BOX{해}}] [eqn: (ln`x) prime][eqn: = lim _{h-> 0} {} {ln`(x+h)`-`ln`x} over {h}]
[eqn: =` lim _{h-> 0} {} {1} over {h} `ln`(1+ {h} over {x} )]
[eqn: = lim _{h -> 0} {} `ln`(1+ {h} over {x} ) ^{{1} over {h}}]
[eqn: = lim _{h -> 0} {} `ln`(1+ {h} over {x} ) ^{{x} over {h} TIMES {1} over {x}}]
[eqn: = lim _{h -> 0} {} `ln`e ^{{1} over {x}}]
[eqn: = {1} over {x} ~]

[eqn: (log` _{a} `x) prime `=` {1} over {x`ln`a}]

[eqn: {BOX{해}}] [eqn: (log` _{a} `x) prime][eqn: =( {ln`x} over {ln`a} )`']
[eqn: = {1} over {x`ln`a}]

[eqn: (sin`x) prime `=`cos`x]

[eqn: {BOX{해}}] [eqn: (sin`x) prime][eqn: = lim _{h-> 0} {} {sin`(x+h)`-`sin`x} over {h}]
[eqn: = lim _{h -> 0} {} {sin`x`cos`h+cos`x`sin`h`-`sin`x} over {h}]
[eqn: = lim _{h -> 0} {} {sin`x(cos`h-1)`+`cos`x`sin`h} over {h}]
[eqn: = lim _{h -> 0} {} `sin`x {h(cos`h-1)} over {h ^{2}} +cos`x` {sin`h} over {h}]
[eqn: =cos`x]

[eqn: (cos`x) prime `=`-sin`x]

[eqn: {BOX{해}}] [eqn: (cos`x) prime][eqn: = lim _{h -> 0} {} {cos`(x+h)`-`cos`x} over {h}]
[eqn: = lim _{h -> 0} {} {cos`x`cos`h`-sin`x`sin`h`-`cos`x} over {h}]
[eqn: = lim _{h -> 0} {} `cos`x {h(cos`h-1)} over {h ^{2}} -sin`x` {sin`h} over {h}]
[eqn: =-sin`x]

[eqn: (tan`x) prime `=`sec ^{2} `x]

[eqn: {BOX{해}}] [eqn: (tan`x) prime][eqn: =( {sin`x} over {cos`x} )']
[eqn: = {cos ^{2} `x`-(-sin ^{2} `x)} over {cos ^{2} `x}]
[eqn: = {1} over {cos ^{2} `x}]
[eqn: =sec ^{2} `x]

[eqn: (cot`x) prime `=`-csc ^{2} `x]

[eqn: {BOX{해}}] [eqn: (cot`x) prime][eqn: =( {cos`x} over {sin`x} ) prime]
[eqn: = {-sin ^{2} `x`-`cos ^{2} `x} over {sin ^{2} `x}]
[eqn: =- {1} over {sin ^{2} `x}]
[eqn: =-csc ^{2} `x]

[eqn: (sec`x) prime `=`sec`x`tan`x]

[eqn: {BOX{해}}] [eqn: (sec`x) prime `][eqn: =( {1} over {cos`x} )']
[eqn: = {sin`x} over {cos ^{2} `x}]
[eqn: = {1} over {cos`x} TIMES {sin`x} over {cos`x}]
[eqn: =sec`x`tan`x]

[eqn: (csc`x) prime `=`-csc`x`cot`x]

[eqn: {BOX{해}}] [eqn: (csc`x) prime `][eqn: =( {1} over {sin`x} ) prime]
[eqn: = {-cos`x} over {sin ^{2} `x}]
[eqn: =- {1} over {sin`x} TIMES {cos`x} over {sin`x}]
[eqn: =-csc`x`cot`x]

[eqn: (f(x)g(x))'`=`f`'(x)g(x)`+`f(x)g`'(x)]

[eqn: {BOX{해}}] [eqn: (f(x)g(x)) prime][eqn: = lim _{h-> 0} {} {f(x+h)g(x+h)-f(x)g(x)} over {h}]
[eqn: = lim _{h -> 0} {} {f(x+h)g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)} over {h}]
[eqn: = lim _{h -> 0} {} {f(x+h)g(x+h)-f(x)g(x+h)} over {h} + {f(x)g(x+h)-f(x)g(x)} over {h}]
[eqn: = lim _{h -> 0} {} `g(x+h) {f(x+h)-f(x)} over {h} +f(x) {g(x+h)-g(x)} over {h}]
[eqn: =f`'(x)g(x)+f(x)g`'(x)]

[eqn: (ln`f(x))'`=` {f`'(x)} over {f`(x)}]

[eqn: {BOX{해}}] [eqn: {d(ln`f(x))} over {dx}][eqn: = {d(ln`t)} over {dx} `````````````````````````````(t=f(x))]
[eqn: = {d(ln`t)} over {dt} TIMES {dt} over {dx}]
[eqn: = {1} over {t} TIMES f`'(x)``````````````````( {dt} over {dx} = {d(f(x))} over {x} =f`'(x))]
[eqn: = {f`'(x)} over {f(x)}]

적분공식

[eqn: int _{} ^{} {k`dx`=`kx`+C}]

[eqn: int _{} ^{} {x ^{n} `dx`=` {1} over {n+1} x ^{n+1} `+C}] (단, [eqn: n` != `-1])

[eqn: int _{} ^{} {f`'(x){f(x)} ^{n} `dx`=` {1} over {n+1} {f(x)} ^{n+1} `+C}] (단, [eqn: n` != `-1])

[eqn: int _{} ^{} {{1} over {x} `dx`=`ln`|x|`+C}]

[eqn: {BOX{해}}] 만약 [eqn: x>0]이라면
[eqn: (ln`x)'= {1} over {x}]
[eqn: THEREFORE int _{} ^{} {{1} over {x} dx`=`ln`x}]

만약 [eqn: x<0]이라면
[eqn: (ln`(-x))'`=` {-1} over {-x} `=` {1} over {x}]
[eqn: THEREFORE int _{} ^{} {{1} over {x} dx`=`ln`(-x)}]

[eqn: int _{} ^{} {{f` prime (x)} over {f(x)} `dx`=`ln``|f(x)|`+C}]

[eqn: int _{} ^{} {(ax+b) ^{n} `dx`=` {1} over {a(n+1)} (ax+b) ^{n+1} `+C}]

[eqn: int _{} ^{} {{c} over {ax+b} `dx`=` {c} over {a} `ln`(ax+b)`+C}]

[eqn: int _{} ^{} {e ^{x} `dx`=`e ^{x} `+C}]

[eqn: int _{} ^{} {a ^{x} `dx`=` {a ^{x}} over {ln`a} `+C}]

[eqn: int _{} ^{} {ln`x`dx`=`x`ln`x`-`x+C}]

[eqn: {BOX{해}}] [eqn: (f(x)g(x))'`=`f`'(x)g(x)+f(x)g`'(x)]이므로
[eqn: f`'(x)g(x)=(f(x)g(x))'`-`f(x)g`'(x)]
[eqn: int _{} ^{} {f`'(x)g(x)`dx`=`f(x)g(x)`-` int _{} ^{} {f(x)g`'(x)`dx}}] [eqn: CDOTS CDOTS]그적미적
따라서
[eqn: int _{} ^{} {ln`x`dx}]
[eqn: = int _{} ^{} {1 TIMES ln`x`dx}]
[eqn: =x`ln`x`-` int _{} ^{} {x TIMES {1} over {x} `dx}]
[eqn: =x`ln`x`-` int _{} ^{} {1`dx}]
[eqn: =x`ln`x`-x+C]

[eqn: int _{} ^{} {log _{a} x`dx`=`x`log _{a} `x`-` {x} over {lna} +C}]

[eqn: {BOX{해}}] [eqn: int _{} ^{} {log _{a} `x} `dx][eqn: = int _{} ^{} {{ln`x} over {lna} `dx}]
[eqn: = {1} over {ln`a} int _{} ^{} {lnx`dx}]
[eqn: = {1} over {ln`a} (x`ln`x-x)`+`C]
[eqn: =x`log _{a} `x- {x} over {ln`a} +C]

[eqn: int _{} ^{} {sin`x`dx`=`-cos`x+C}]

[eqn: int _{} ^{} {cos`x`dx`=`sin`x+C}]

[eqn: int _{} ^{} {tan`x`dx`=`ln`□sec`x□+C}]

[eqn: int _{} ^{} {cot`x`dx`=`ln`□sin`x□+C}]

[eqn: int _{} ^{} {sec ^{2} `x`dx`=`tan`x+C}]

[eqn: int _{} ^{} {csc ^{2} `x`dx`=`-cot`x+C}]

[eqn: int _{} ^{} {sec`x`tan`x`dx`=`sec`x+C}]

[eqn: int _{} ^{} {csc`x`cot`x`dx`=`-csc`x+C}]

그적미적

미      적(분)

<로다삼지>

그항각수

[eqn: {BOX{예}}] [eqn: int _{} ^{} {x`sin`x`dx}][eqn: =x`(-cos`x)`-` int _{} ^{} {-cos`x`dx}]
[eqn: =-x`cos`x`+` int _{} ^{} {cos`x`dx}]
[eqn: =-x`cos`x+sin`x+C]