Calculus
Calculus
Aaron1. Fundamental Theorem of Calculus
\[ \begin{align*} \int_a^b{f'(x)} \,dx & = f(b) - f(a) \end{align*} \]
2. Leibniz Rule
\[ \begin{align*} \frac{d}{dx}\left(\int_a^x f(t) \, dt \right) & = f(x) \end{align*} \]
2.1 Proof
\[ \begin{align*} F(x) & = \lim_{h \to 0} \frac{\int_c^{x+h} f(t) \, dt - \int_c^x f(t) \, dt}{h} \\ & = \lim_{h \to 0} \frac{\int_x^{x+h} f(t) \, dt}{h} \\ & = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h} \\ & = f'(x) \end{align*} \]
3. Product Rule
\[ \begin{align*} \frac{d}{dx}(uv) & = u \frac{dv}{dx} + v \frac{du}{dx} \end{align*} \]
3.1 Proof
\[ \begin{align*} \frac{d}{dx}(uv) & = \lim_{h \to 0} \frac{u(x+h)v(x+h) - u(x)v(x)}{h} \\ & = \lim_{h \to 0} \frac{u(x+h)v(x+h) - u(x+h)v(x) + u(x+h)v(x) - u(x)v(x)}{h} \\ & = \lim_{h \to 0} \left( \frac{u(x+h)(v(x+h) - v(x))}{h} + \frac{v(x)(u(x+h) - u(x))}{h} \right) \\ & = \frac{du}{dx}v(x) + u(x)\frac{dv}{dx} \\ & = u \frac{dv}{dx} + v \frac{du}{dx} \end{align*} \]
4. L'Hôpital's Rule 洛必达法则
\[ \begin{align*} \lim_{x \to a} \frac{f(x)}{g(x)} & = \lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)} \\ & = \lim_{x \to a} \frac{f'(x)}{g'(x)} \\ & = \frac{f'(a)}{g'(a)} \end{align*} \]
