2.1 Derivatives and Rates Change

1. 定義 : 切線 (tangent line)

     曲線 \(y = f(x)\) 在點 \(P(a, f(a))\) 之斜率 (slope) 為

\[ m=\lim_{x \to a} \frac {f(x)-f(a)}{x-a} \]

      (假設極限存在)


      Secant Lines, Tangent Lines, and Limit Definition of a Derivative
      https://clas.sa.ucsb.edu/staff/lee/secant,%20tangent,%20and%20derivatives.htm


2. 瞬時速度 (instantaneous velocity)

\[ v(a)=\lim_{h \to 0} \frac {f(a+h)-f(a)}{h} \]

3. 導數 (derivative)

    \(f(x)\) 在 \(a\) 的導數，記為\(f'(a)\)

\[ f'(a)=\lim_{h \to 0} \frac {f(a+h)-f(a)}{h} \]

\[ f'(a)=\lim_{x \to a} \frac {f(x)-f(a)}{x-a} \]

2.2 The Derivative as a Function

1. \(f(x)\) 在 \(a\) 的導數

\[ f'(a)=\lim_{h \to 0} \frac {f(a+h)-f(a)}{h} \]

2. \(f(x)\) 在 \(x\) 的導數

\[ f'(x)=\lim_{h \to 0} \frac {f(x+h)-f(x)}{h} \]

     導數表示方式

\[ f'(x)=y'=\frac {dy}{dx}=\frac {df}{dx}=\frac {d}{dx}f(x)=Df(x)=D_xf(x) \]

3. 定義

     若\(\,f'(a)\,\)存在，函數\(\,f\,\)在\(\,a\,\)可微 (differentiable)


4. 定理

     若\(\,f\,\)在\(\,a\,\)可微，則\(\,f\,\)在\(\,a\,\)連續


2.3 Differentiation Formulas

1. 常數規則 (constant rule)

\[ \frac {d}{dx}(c)=0 \]

2. 冪次規則 (power rule)

     若 \(n \in N\)，則

\[ \frac {d}{dx}(x^n)=nx^{n-1} \]

3. 純量積規則 (constant multiple rule)

     若 \(\,c\,\)為常數和\(\,f\,\)為可微函數 (differentiable function)，則

\[ \frac {d}{dx}[cf(x)]=cf'(x) \]

4. 加減法規則 (sum and difference rules)

     若 \(\,f\,\)和\(\,g\,\)為可微函數，則

\[ \frac {d}{dx}\left[f(x)\,\pm\,g(x)\right]=f'(x)\,\pm\,g'(x) \]

5. 乘法規則 (product rules)

     若 \(\,f\,\)和\(\,g\,\)為可微函數，則

\[ \frac {d}{dx}\left[f(x)g(x)\right]={\color{red}{f'(x)}}g(x)+f(x){\color{red}{g'(x)}} \]

6. 除法規則 (quotient rules)

     若 \(\,f\,\)和\(\,g\,\)為可微函數，則

\[ \frac {d}{dx}\left[\frac {f(x)}{g(x)}\right]=\frac {{\color{red}{f'(x)}}g(x)-f(x){\color{red}{g'(x)}}}{\left[ {g(x)}\right]^2} \]

7. 廣義冪次規則 (general power rule)

     若 \(n \in R\)，則

\[ \frac {d}{dx}(x^n)=nx^{n-1} \]

2.4 Derivatives of Trigonometric Functions

1. 三角函數的導數 (derivatives of trigonometric functions)

     \((1)\quad\frac {d}{dx}\sin x=\cos x\)
     \((2)\quad\frac {d}{dx}\cos x =-\sin x\)
     \((3)\quad\frac {d}{dx}\tan x=\sec^2x\)
     \((4)\quad\frac {d}{dx}\cot x=-\csc^2x\)
     \((5)\quad\frac {d}{dx}\sec x=\sec x\tan x\)
     \((6)\quad\frac {d}{dx}\csc x=-\csc x\cot x\)


     Cuemath Derivatives https://www.cuemath.com/calculus/derivatives/


2. 三角函數的極限 (trigonometric limits)

\[\color{red}{\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1} \]
\[\color{red}{\lim_{\theta \to 0} \frac{\cos \theta-1}{\theta} = 0} \]

2.5 The Chain Rule

1. 連鎖律 (chain rule)

     若\(\,g\,\)在\(\,x\,\)處可微,\(\,f\,\)在\(\,g(x)\,\)處可微，則合成函數\(F=\,f \circ g\,\) 在 x 處可微，
     其\(\,F(x) =f(g(x))\,\)其微分為

\[F'(x)=f'(g(x))\cdot g'(x)\]

2. 萊布尼茲表示法 (Leibniz notation)

     若\(\,y=f(u)\,\)和\(\,u=g(x)\,\)處可微函數，則\[\frac {dy}{dx}=\frac {dy}{du}\frac {du}{dx}\]


2. 萊布尼茲表示法 (Leibniz notation)

     若\(\,y=f(u)\,\)和\(\,u=g(x)\,\)處可微函數，則\[\frac {dy}{dx}=\frac {dy}{du}\frac {du}{dx}\]

3. 廣義冪次規則 (general power rule)

     若 \(\,n \in R\,\)和\(\,u=g(x)\,\)，則

\[ \frac {d}{dx}(u^n)=nu^{n-1}\frac {du}{dx} \]

     或

\[ \frac {d}{dx}[g(x)]^n=n[g(x)]^{n-1}g'(x) \]

2.6 Implicit Differentiation

隱函數微分 (Implicit Differentiation) Math’s Fun Advance https://www.mathsisfun.com/calculus/implicit-differentiation.html


Reference
Essential Calculus, metric edition 2e, (2022) James Stewart, Daniel K. Clegg, Saleem Watson, Cengage Learning.