AP Calculus AB/BC - Unit 2: Differentiation

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Definition of Derivative:

$f'(c) = \lim_{{x \to c}} \frac{f(x) - f(c)}{x - c}$

Basic Derivatives:

$\frac{d}{dx} (x^n) = nx^{n-1}$
$\frac{d}{dx} (\sin x) = \cos x$
$\frac{d}{dx} (\cos x) = -\sin x$
$\frac{d}{dx} (\tan x) = \sec^2 x$
$\frac{d}{dx} (\cot x) = -\csc^2 x$
$\frac{d}{dx} (\sec x) = \sec x \tan x$
$\frac{d}{dx} (\csc x) = -\csc x \cot x$
$\frac{d}{dx} (\ln u) = \frac{1}{u} \frac{du}{dx}$
$\frac{d}{dx} (e^u) = e^u \frac{du}{dx}$

Differentiation Rules:

Chain Rule:

$\frac{d}{dx} [f(u)] = f'(u) \frac{du}{dx} \quad \text{OR} \quad \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Product Rule:

$\frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx} \quad \text{OR} \quad u v' + v u'$

Quotient Rule:

$\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \quad \text{OR} \quad \frac{v u' - u v'}{v^2}$

Theorems of Differentiation:

Intermediate Value Theorem:

If the function $f(x)$ is continuous on $[a, b]$ , and $y$ is a number between $f(a)$ and $f(b)$ , then there exists at least one number $x = c$ in the open interval $(a, b)$ such that $f(c) = y$ .

Mean Value Theorem:

If the function $f(x)$ is continuous on $[a, b]$ AND the first derivative exists on the interval $(a, b)$ , then there is at least one number $x = c$ in $(a, b)$ such that:

$f'(c) = \frac{f(b) - f(a)}{b-a}$

Rolle's Theorem:

If the function $f(x)$ is continuous on $[a, b]$ , AND the first derivative exists on the interval $(a, b)$ AND $f(a) = f(b)$ , then there is at least one number $x = c$ in $(a, b)$ such that:

$f'(c) = 0$

Extreme Value Theorem:

If the function $f(x)$ is continuous on $[a, b]$ , then the function is guaranteed to have an absolute maximum and an absolute minimum on the interval.