Average rate of change over [a, b]: $\frac{f(b) - f(a)}{b - a}$

Instantaneous rate of change at x = a: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Alternative form: $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Derivative Definition f'(x)

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

Notation:

• f'(x): Lagrange notation
• y': Newton notation
• $\frac{dy}{dx}$: Leibniz notation
• $\dot{y}$: Newton's dot notation (for time derivatives)

Numerical approximation: $f'(a) \approx \frac{f(a+h) - f(a-h)}{2h}$

More accurate than one-sided difference quotient.

Given table of values, use average rate of change over smallest interval:

f'(1.1) ≈ (3.0 - 2.5) / (1.2 - 1.0) = 0.5 / 0.2 = 2.5

A function is differentiable at c if f'(c) exists.

Non-differentiable points:

• Corner: Different left/right derivatives
• Cusp: Derivatives approach ±∞ from each side
• Discontinuity: Jump, removable, or infinite
• Vertical tangent: Derivative approaches infinity

Example: f(x) = |x| has a corner at x = 0 (not differentiable there, but continuous).

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

Derivative of x^n

Works for all real numbers n:

• $\frac{d}{dx}(x^5) = 5x^4$
• $\frac{d}{dx}(x^{-3}) = -3x^{-4} = -\frac{3}{x^4}$
• $\frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$
• $\frac{d}{dx}(1) = 0$ (since 1 = x0)

• Constant: $\frac{d}{dx}(c) = 0$
• Constant multiple: $\frac{d}{dx}[cf(x)] = cf'(x)$
• Sum: $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$
• Difference: $\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$

Derivative is a linear operator: $\frac{d}{dx}[af(x) + bg(x)] = af'(x) + bg'(x)$

$\frac{d}{dx}(\sin x) = \cos x$ $\frac{d}{dx}(\cos x) = -\sin x$ $\frac{d}{dx}(e^x) = e^x$ $\frac{d}{dx}(\ln x) = \frac{1}{x}, \quad x > 0$

Special limits (used to derive these): $\lim_{x \to 0}\frac{\sin x}{x} = 1, \quad \lim_{x \to 0}\frac{1 - \cos x}{x} = 0$

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

Mnemonic: "first d-second plus second d-first"

Example: $\frac{d}{dx}(x^2 \sin x) = 2x \cdot \sin x + x^2 \cdot \cos x$

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

Mnemonic: "low d-high minus high d-low, over the square of what's below"

Example: $\frac{d}{dx}\left(\frac{x}{x+1}\right) = \frac{1 \cdot (x+1) - x \cdot 1}{(x+1)^2} = \frac{1}{(x+1)^2}$

Derivatives of tan x, cot x, sec x, csc x

$\frac{d}{dx}(\tan x) = \sec^2 x$ $\frac{d}{dx}(\cot x) = -\csc^2 x$ $\frac{d}{dx}(\sec x) = \sec x \tan x$ $\frac{d}{dx}(\csc x) = -\csc x \cot x$

Derived using quotient rule on sin/cos.