Definition of derivative

\[f' = \frac{d}{dx}f(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}\] \[\frac{d}{dx}cx^n = cn x^{n-1}\]

exponential and logarithm functions

\[\begin{align} & \frac{d}{dx} e^x & = & \ e^x \\ & \frac{d}{dx} a^x & = & \ a^x \ln a \\ & \frac{d}{dx} \ln x & = & \ \frac{1}{x} \\ & \frac{d}{dx} \log_a x & = & \ \frac{1}{x \ln a} \end{align}\]

Chain Rule

\[F'(x) = f' \left( g(x) \right) g'(x)\]

L’Hospital’s Rule

\[\text{在以下状况中: (a可为real number, 正负infinity) } \\ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0} \text{ 或 } \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\pm \infty}{\pm \infty} \\ \text{可得} \\ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\]

Derivatives of Functions

  • Add and Subtract Derivatives of Functions

    \[(f+g)' = f' + g'\] \[\frac{d}{dx}[f(x)-g(x)] = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)\]

  • Derivatives of Products and Powers of Functions

    \[(fg)' = f'g + fg'\]

  • Derivatives of Quotients of Functions

    \[(\frac{f}{g})' = \frac{f'g - fg'}{g^2}\]

  • The Chain Rule for Differentiating Complicated Functions

    \[\frac{d}{dx}f(u(x)) = \frac{d}{du}f(u) \times \frac{d}{dx}u(x)\]

  • for a Point(x,y)

    \[Point(x,y), x=f(t), y=g(t)\] \[\frac{dy}{dx} = \frac{dy}{dt}\frac{dt}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\]

Derivatives of six trig functions

base Facts

\[\lim_{\theta \rightarrow 0} \frac{\sin \theta}{\theta} = 1 \\ \lim_{\theta \rightarrow 0} \frac{\cos \theta - 1}{\theta} = 0\]

derivatives for 6 trigonometry:

\[\begin{align} \frac{d}{dx} \sin(x) & = & & \cos(x) \\ \frac{d}{dx} \cos(x) & = & - & \sin(x) \\ \frac{d}{dx} \tan(x) & = & & \sec^2(x) \\ \frac{d}{dx} \cot(x) & = & - & \csc^2(x) \\ \frac{d}{dx} \sec(x) & = & & \sec(x) \ tan(x) \\ \frac{d}{dx} \csc(x) & = & - & \csc(x) \ cot(x) \end{align}\]