Chapter 3: Derivatives Foundations
A structured introduction from rate-of-change meaning to classical rule systems and applied optimization.
3.1 Why Derivatives
A derivative measures instantaneous change.
- Geometry: derivative is slope of the tangent line.
- Physics: derivative measures velocity and acceleration.
- Modeling: derivative tracks growth, decay, and sensitivity.
Average rate on $[a,b]$:
$$\frac{f(b)-f(a)}{b-a}$$
Instantaneous rate at $x$:
$$f'(x)$$
3.2 Limit Definition
$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$
- Key idea. This limit is the source of all derivative rules.
- Key idea. If the limit does not exist, derivative does not exist at that point.
Example 3.2.1
For $f(x)=x^2$:
$$\frac{f(x+h)-f(x)}{h}=\frac{(x+h)^2-x^2}{h}=2x+h$$
As $h\to0$, $f'(x)=2x$.
3.3 Core Rules
$$\frac{d}{dx}[c]=0,\quad \frac{d}{dx}[x^n]=nx^{n-1},\quad \frac{d}{dx}[f\pm g]=f'\pm g'$$
Example 3.3.1
$y=5x^4-3x^2+7x-9$
$$y'=20x^3-6x+7$$
3.4 Product, Quotient, and Chain
$$\frac{d}{dx}[uv]=u'v+uv'$$
$$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2}$$
$$\frac{d}{dx}[f(g(x))]=f'(g(x))g'(x)$$
- Key idea. Chain rule is mandatory for compositions (inside function).
3.5 Applications
Motion
If $s(t)$ is position:
$$v(t)=s'(t),\quad a(t)=s''(t)$$
Optimization
Construct objective function, compute derivative, solve $f'(x)=0$, and classify extrema.
3.6 Progressive Practice
1. Differentiate $7x^5-2x+1$.
2. Differentiate $(3x^2+4)^6$.
3. Differentiate $x^2\\sin x$.
4. Find $dy/dx$ if $x^2+y^2=16$.