Calculus Basics
Limits
The limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ gets arbitrarily close to.
$$ \lim_{x \to a} f(x) = L $$
Formal Definition
For every $\varepsilon > 0$, there exists a $\delta > 0$ such that:
$$ 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon $$
Derivatives
The derivative of $f$ at $x$ is defined as the limit:
$$ f’(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$
Power Rule
$$ \frac{d}{dx} x^n = n x^{n-1} $$
Product Rule
If $u(x)$ and $v(x)$ are differentiable, then:
$$ \frac{d}{dx}[u(x) \cdot v(x)] = u’(x)v(x) + u(x)v’(x) $$
Chain Rule
$$ \frac{d}{dx} f(g(x)) = f’(g(x)) \cdot g’(x) $$
Common Derivatives
| $f(x)$ | $f’(x)$ |
|---|---|
| $x^n$ | $n x^{n-1}$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $e^x$ | $e^x$ |
| $\ln x$ | $\frac{1}{x}$ |
Integrals
The definite integral of $f$ from $a$ to $b$:
$$ \int_a^b f(x),dx = F(b) - F(a) $$
where $F’(x) = f(x)$.
Fundamental Theorem of Calculus
$$ \frac{d}{dx} \int_a^x f(t),dt = f(x) $$
Practice Problem
Evaluate the limit:
$$ \lim_{x \to 0} \frac{\sin x}{x} $$
Answer: This famous limit equals $1$. It can be proven using the squeeze theorem with the inequality $\cos x \leq \frac{\sin x}{x} \leq 1$ for small $x$.