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

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

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

This represents the slope of the tangent line at point x.

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

The derivative at a point equals the slope of the tangent line to the curve at that point. Geometrically, it represents the instantaneous rate of change.

If f is differentiable at x = c, then f is continuous at x = c.

Converse is false: Continuous does NOT guarantee differentiable (e.g., |x| at x = 0 has a corner).

A function may not be differentiable at points where:

• Corner: Different left and right derivatives
• Cusp: Derivative approaches ±∞
• Discontinuity: Jump, removable, or infinite
• Vertical tangent: Derivative undefined

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

Works for all real numbers n.

Examples:

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

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

$\frac{d}{dx}(e^x) = e^x$

Derivative of general exponential: $\frac{d}{dax}(a^x) = a^x \ln a$

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

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

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

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