6 Basic Derivatives
6.1 Summary of Derivatives
Derivatives quantify how a function changes with respect to its input, capturing slopes, rates of change, and tangent behavior. They play a central role in calculus by enabling the analysis of motion, growth, optimization, and a wide range of dynamic processes. Key concepts, descriptions, and applications are summarized in Table 6.1.
| KeyConcept | Description | ExampleApplication |
|---|---|---|
| Definition | \(f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\); rate of change at \(x\) | Instantaneous velocity: \(v(t) = s'(t)\) |
| Interpretation | Slope of tangent line; instantaneous rate of change | Slope of a hill: \(m = h'(x)\) |
| Basic Rules | Power rule, sum rule, constant multiple rule | \(\frac{d}{dx} x^n = n x^{n-1}\), \(\frac{d}{dx}[f+g] = f' + g'\) |