derivative
π ββοΈν΄λν°μΌλ‘ λ³Ό λ νΉμ κΈμλ μ«μκ° νλ©΄μ λ€ μλμ€λ©΄, ν΄λν° κ°λ‘λ‘ λ리μλ©΄ λ©λλ€
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<λͺ©μ°¨>
0. μμμΌ ν κ²
1. μΌκ°ν¨μμ λν¨μ
1-1 (1)μ μ¦λͺ
1-2 (2)μ μ¦λͺ
1-3 (3)μ μ¦λͺ
1-4 (4)μ μ¦λͺ
1-5 (5)μ μ¦λͺ
1-6 (6)μ μ¦λͺ
1-7 μμ λ―ΈλΆν κ²°κ³Όλ€μ μ λΆνλ©΄?
2. μ§μλ‘κ·Έν¨μμ λν¨μ
2-1. (2) μ¦λͺ
2-2. (1) μ¦λͺ
2-3. (4) μ¦λͺ
2-4. (3) μ¦λͺ
3. μλμ κ°μλ
4. λ―ΈλΆ κ°λ¨ν μμ
5. μ λΆ κ°λ¨ν μμ
0. μμμΌ ν κ²
β»λ‘κ·Έμ μ§μλ νμ +(μ)λΆνΈλ€
β»λͺ«μ λ―ΈλΆ (red)(λ§€μ° μ€μνλ€)
\(\left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{ \left\{ g(x) \right\}^2 }\)
ex) \(\left( \frac{1}{x} \right)' = \frac{0 \cdot x -1 \cdot 1}{x^2} = \frac{-1}{x^2}\)
1. μΌκ°ν¨μμ λν¨μ
(1) \((sinx)'= cosx\)
(2) \((cosx)' = -sinx\)
(3) \((tanx)' = sec^2 x\)
(4) \((cotx)'=-csc^2 x\)
(5) \((secx)' = secx \cdot tanx\)
(6) \((cscx)' = -cscx \cdot cotx\)
μ΄κ±° μ½κ² μΈμ°λ λ² \(\color{red}{\Rightarrow}\) cλ‘ μμνλλ°μ λ―ΈλΆνλ κ²μ -
λΆνΈκ° λΆλ€
1-1. (1)μ μ¦λͺ
\(f(x)=sinx\)
\((sinx)' = f'(x)=\lim_{h\to0} \frac{f(x+h)-f(x)}{h} = \lim_{h\to0} \frac{sin(x+h)-sin(x)}{h}\)
μ¬κΈ°μ x+h \(\color{red}{\Rightarrow}\) A, Β Β Β x \(\color{blue}{\Rightarrow}\) BλΌ μ€μ νκ² λ€
μΌκ°ν¨μ λ§μ
곡μμ μ¬μ©νμ. μλλ $sin(A-B)$μμ μμΉλ§ μ‘°κΈ λ°κΎΌ κ²μ΄λ€ κ²°κ³Όλ κ°λ€
πΌμ°Έκ³ \(sin(A)-sin(B)=2cos \left( \frac{A+B}{2} \right) sin \left( \frac{A-B}{2} \right)\)
κ·ΈλΌ μμμ μ νA, Bλ₯Ό μ¬κΈ°μ λμ
ν΄λ³΄λ©΄ μ΄λ€ μμ΄ λ μ°μΆλμ§?
μμ μΌκ°ν¨μ sin곡μμ μ΄μ©νμ¬ \(\lim_{h\to0} \frac{sin(x+h)-sin(x)}{h}\)μ λ€μ μ κ°νλ©΄ μλμ κ°λ€
\(\lim_{h\to0} \frac{2cos(x+\frac{h}{2}) \cdot sin(\frac{h}{2})}{h} = \frac{cos(x+\frac{h}{2}) \cdot sin(\frac{h}{2})}{\frac{h}{2}}\)
μ΄λ κ² λ³νμν€λ©΄ μ°μΈ‘ κ·Ήνκ°μΈ sinμͺ½μ 1λ‘ μλ ΄νκ³ ,
μ’μΈ‘ κ·Ήνκ°μΈ cosμͺ½μ \(cosx\) λ‘ μλ ΄νλ€
1-2. (2)μ μ¦λͺ
\(f(x) = cosx\)
\((cosx)'=f'(x)=\lim_{h\to0} \frac{cos(x+h) - cos(x)}{h}\)
μ¬κΈ°μ 2x+hλ₯Ό A+B, Β Β hλ₯Ό A-Bλ‘ λ³΄μ
κ·Έλ¦¬κ³ μλμ μ°Έκ³
\(cos(A)-cos(B) \\ =-2sin \left( \frac{A+B}{2} \right) sin \left( \frac{A-B}{2} \right)\)
μ΄μ΄μ μμ μμ λμ
νλ©΄ \(\lim_{h\to0} \frac{-2sin(x+\frac{h}{2}) \cdot sin(\frac{h}{2})}{h} = \lim_{h\to0} \frac{-sin(x+\frac{h}{2}) \cdot sin(\frac{h}{2})}{\frac{h}{2}}\)
μ΄λ κ² λλ©΄ μ λΆ μλ ΄νκ³ λ¨λ κ²μ \(-(sinx \cdot 1) \cdot 1 \Rightarrow -sinx\)
1-3. (3)μ μ¦λͺ
\(f(x) = tan(x)\)
\(=\left( \frac{sin(x)}{cos(x)} \right)' = \frac{(sinx)\ \cdot cosx - sinx \cdot (cosx)'}{cos^2 x} = \frac{cos^2 x + sin^2 x}{cos^2 x} = \frac{1}{cos^2 x} \\ \therefore sec^2 x\)
1-4. (4)μ μ¦λͺ
\(f(x) = (cotx)'\)
μ cotμ΄ λλ? Β λ°λ‘ \(\frac{1}{tanx}\)μ§ μλλ
\(\color{pink}{\Rightarrow}\) \(\frac{(-sinx) \cdot sinx - cosx \cdot cosx}{sin^2 x} = \frac{-1}{sin^2 x} = -csc^2 x\)
1-5. (5)μ μ¦λͺ
\(f(x)= (secx)'\)
\(=\left( \frac{1}{cos(x)} \right)' = \frac{0 \cdot cosx -1 \cdot(-sinx)}{cos^2 x} = \frac{1}{cosx} \cdot \frac{sinx}{cosx} = secx \cdot tanx\)
β»μΌκ°ν¨μλ₯Ό λ―ΈλΆνμ λ κ°μ κ·Έλλ‘ λμ¨λ€! Β Β ex) \(3x\)
ex-1) \((sec3x)' = 3 \cdot sec3x \cdot tan3 x\)
ex-2) \((tan6x)' =6 \cdot sec^2 6x\)
1-6. (6)μ μ¦λͺ
\(f(x)= (csc(x))'\)
λ°κΏμ°λ©΄ \(\left( \frac{1}{sin(x)} \right)'\) μ΄λ κ² λλλ° μ¬κΈ°μ ν©μ±ν¨μ λ―ΈλΆκ³΅μ μ°μ \(\left( \frac{f(x)}{g(x)} \right)'\)
\(\Rightarrow\) \(\frac{0x \cdot sinx - 1 \cdot cosx}{sin^2x} = \frac{-1}{sinx} \cdot \frac{cosx}{sinx} = -cscx \cdot cotx\)
1-7. μμ λ―ΈλΆν κ²°κ³Όλ€μ μ λΆνλ©΄?
(1) \(\int cosx dx = sinx + C\)
(2) \(\int sinx dx = -cosx + C\)
(3) \(\int sec^2xdx = tanx + C\)
(4) \(\int csc^2xdx = -cotx+C\)
(5) \(\int secx \cdot tanxdc = secx+C\)
(6) \(\int cscx \cdot cotxdx = -csc + C\)
2. μ§μ $\cdot$ λ‘κ·Έν¨μμ λν¨μ
(1) \(\left( e^x \right)' = e^x\)
(2) \(\left( a^x \right)' = a^x lna\)
(3) \(\left( lnx \right)' = \frac{1}{x}\)
(4) \(\left( log_ax \right)' = \frac{1}{x} \cdot \frac{1}{lna}\)
μλμ μ΄ν΄νκΈ° μ½κ² (2) β (1) β (4) β (3) μμΌλ‘ μ¦λͺ
νκ² λ€
2-1. (2) μ¦λͺ
\(f'(x) = \left( a^x \right)' = \lim_{h\to0} \frac{f(x+h)-f(x)}{h} = lim_{h\to0} \frac{a^{x+h}-a^x}{h}\)
π²μ°Έκ³
\(lim_{x\to0} \frac{a^{px}-1}{qx} = \frac{p}{q} \cdot lna\)
\(lim_{x\to0} \frac{a^{x}-1}{x} = lna\)
μ΄κ±Έ μ°Έκ³ ν΄μ μ΄μ΄μ μ κ°νλ©΄
\(\lim_{h\to0} \frac{a^h-1}{h} \cdot a^x \Rightarrow a^x \cdot lna\)
\(\color{red}{ex)}\)
\(\left( 3^x \right)' = 3^x \cdot ln3\)
\(\left( 3^{2x} \right)' = \frac{2}{1} \cdot 3^{2x} \cdot ln3\)
2-2. (1) μ¦λͺ
\(\left( e^{x} \right)' = e^x \cdot ln_e e = e^x\)
\(\color{red}{ex)}\)
\(\left( e^{3x} \right)' = 3e^{3x} \cdot ln_e e = 3e^{3x}\)
2-3. (4) μ¦λͺ
\(f(x)=log_ax\) λΌκ³ νμ
\(\left( log_ax \right)' = f'(x) = lim_{h\to0} \frac{log_a(x+h) - log_a x}{h}\) μ¬κΈ°μ λΆλͺ¨λ₯Ό μμ κ³ λΆμλ§ λ³΄μ
ποΈμ κΉ λ°μ΄ κ°μ λ‘κ·Έλ₯Ό λΉΌλ©΄? \(\color{red}{\Rightarrow}\) \(log_a A- log_b B = log_a \frac{A}{B}\)
μ μ΄λ₯Ό νμ©νλ©΄ \(log_a\left(\frac{x+h}{x} \right)\)λ‘ λνλΌ μ μλ€? Β μ κ·Έλ¬λ©΄ μλμμλ xλ₯Ό λλκ³ μ΄μ΄μ μ κ°νκ² λ€
\(= \lim_{h\to0} \frac{1}{h} \cdot log_a (1+\frac{h}{x})^{\frac{1}{h}}\) (red) \(=\) \(\lim_{h\to0} log_a (1+\frac{h}{x})^{\frac{x}{h} \cdot \frac{1}{x}}\)
μ΄? κ°λ§λ³΄λ \((1+\frac{h}{x})^{\frac{x}{h}} = e\) λ€? \(log_a e^{\frac{1}{x}} = \frac{1}{x}log_a e = \frac{1}{x}\cdot\frac{1}{log_e a}\)
\(\color{purple}{\Rightarrow}\) Β \(\frac{1}{x} \cdot \frac{1}{lna}\)
βμ΄κ²λ μ°Έκ³ νμ
ex-1) \(log_a s = \frac{1}{log_s a}\)
ex-2) \(\left(log_2 x \right)' = \frac{1}{x} \cdot \frac{1}{ln2}\)
ex-3) \(\left( log_2 3x \right)' = 3 \cdot \frac{1}{3x}\cdot \frac{1}{1n2}\)
ex-4) \(\left( log_a f(x) \right)' = \frac{f'(x)}{f(x)} \cdot \frac{1}{lna}\)
ex-5) \(\left( log_2 3^{2x} \right)' = \left( 2x \cdot log_2 3 \right)'\) \(\color{pink}{\Rightarrow}\) \(2 \cdot log_2 3\)
2-4. (3) μ¦λͺ
\(f'\left( x \right) = \left( log_e x \right)' = \frac{1}{x} \cdot \frac{1}{ln_e e} = \frac{1}{x}\)
ex) \(ln(\Delta)' = \frac{\Delta'}{\Delta}\) \(\color{pink}{\Rightarrow}\) \(\{ ln(x^2+x+1) \}' = \frac{\{2x+1\} \cdot ln_e e}{\{x^2+x+1\} \cdot ln_e e} = \frac{2x+1}{x^2+x+1}\)
3. μλμ κ°μλ
λ©μ΄νμ€ν 리μ μμΌλ보μ΄κ° μμ§μΈλ€κ³ μκ°ν΄λ³΄μ
μλλ CartesianΒ coordinate systemμμ \((x+k)x^2\) μ λνλΈ κ²μ΄λ€
μλλ μμΌλ 보μ΄κ° λλ¬νλ ꡬκ°λ€μ μμλ‘ xμΆ, yμΆμ μ΄μ©ν΄ λνλΈ κ²μ΄λ€
μΌλ°μ μΌλ‘ μ’ννλ©΄ μλ₯Ό μμ§μ΄λ μ Pμ t(μκ°)μμμ μμΉ(x,y)λ
tλ₯Ό 맀κ°λ³μλ‘ νλ λ ν¨μ \(x=f(t)\), Β \(y=g(t)\)λ‘ λνλΌ μ μλ€
μ°Έκ³ λ‘ \(`t(μκ°) >0`\) λΉμ°ν κ²μ΄λ€
λ€μμ μμΌλ보μ΄κ° \(p_1(1, 6)\)μμ \(P_?(2, 17)\)κΉμ§ μ΄λνμ λ
tμ΄ κ±Έλ¦° κ²μ λν κ·Έλνλ₯Ό λνλΈ κ²μ΄λ€.
μ΄λ μ Pμ t(μκ°)μμμ μλμ μλ ₯, κ°μλμ κ°μλμ ν¬κΈ°λ λ€μκ³Ό κ°λ€
(1) μ Pμμ t(μκ°)μμμ μλμ μλ ₯
(2) μλ: ( \(f'(t), g'(t)\) )
(3) μλ ₯: \(\sqrt{f'(t)^2 + g'(t)^2}\)
*거리: Β \(t \cdot μλ ₯\)
μκ°μ μΈ xμΆκ³Ό yμΆμ λ³νμ¨μ λν΄ \(\frac{dx}{dt} = f'(t)\)μ \(\frac{dy}{dt} = g'(t)\) λ‘ νννλ€.
μ κ·Έλ¬λ©΄ \(p_1 \rightarrow p_?\) ꡬκ°μ μ§λκ° λ 3μ΄ μΌλμ μλ ₯μ μ΄λ¨μ§ κ³μ°ν΄λ³΄μ
νΌνκ³ λΌμ€ μ 리λ₯Ό μ°μνλ©΄ μ½λ€
μ°μ λμ μ¬μ΄ 거리 ꡬνλ 곡μμΈ μ ν΄λ¦¬λ 거리 곡μ\(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) μ μ΄μ©ν΄μ
\(\sqrt{122}\)κ° λμκ³ κ·Έ ꡬκ°μμμ 3μ΄κ° κ²½κ³Όνμ λ μλ ₯μ \(\frac{\sqrt{122}}{3}\)μ΄λ€
κ·Έλ λ€λ©΄ κ°μλ ₯μ μ΄λ»κ²??
(λ―ΈλΆ 2λ²μ΄λ€)
(1) κ°μλ: \((f''(t), g''(t))\)
(2) κ°μλμ ν¬κΈ°: \(\sqrt{f''(t)^2 + g''(t)^2}\)
4. λ―ΈλΆ κ°λ¨ν μμ
ν¨μ±ν¨μμ λ―ΈλΆ: \((\clubsuit \cdot \Delta)'\) \(\color{pink}{\Rightarrow}\) \(\clubsuit' \cdot \Delta + \clubsuit \cdot \Delta'\)
ex-1) \(f(x) = e^x \cdot sin5x\)
\(f'(x) = e^x sin5x + e^x (5cos5x)\)
ex-2) \(f(x) = e^{-2x}cos3x\)
\(f'(x) = (-2 \cdot e^{-2x})cos3x + e^{-2x}(-3 \cdot sin3x) \\ = -e^{-2x}(2 \cdot cos3x + 3 \cdot sin3x)\)
5. μ λΆ κ°λ¨ν μμ
ex-1) \(\{ ln(x^2+1) \}' = \frac{2x}{x^2+1}\) μ΄κ²μ λ―ΈλΆλ κ°μ΄λ€
κ·ΈλΌ μ λΆμ? Β \(\int \frac{2x}{x^2+1}dx = ln(x^2+1) + C\)
ex-2) \(\int tanx dx\)
\(= \int \frac{sinx}{cosx}dx = -\int \frac{-sinx}{cosx}dx \Rightarrow -ln |cosx| +C\)
ex-3) $\int e^{3x} dx = \frac{1}{3}e^{3x} + C$
ex-4) $\int cos3xdx = \frac{1}{3} sin3x + C$
μ°Έκ³
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