Postgraduate Science

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Derivatives & Differentiation

The derivative as a limit, geometric interpretation, and the basic rules.

The derivative of $f$ at $a$ is

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

when this limit exists. Geometrically it is the slope of the tangent line to the graph at $x=a$. Differentiability implies continuity (not conversely — $|x|$ at $0$).

Core rules:

$$(f+g)' = f' + g', \quad (fg)' = f'g + fg', \quad (f \circ g)'(x) = f'(g(x))\, g'(x).$$

For functions of several variables, partial derivatives $\partial f/\partial x_i$ and the total derivative (Jacobian matrix) generalize the idea. The gradient $\nabla f$ assembles the partials into a vector pointing in the direction of steepest ascent.

Interactive: tangent line at a point

Quiz

1. The derivative $f'(a)$ is geometrically:
2. Differentiability at a point implies:
3. $(fg)'$ equals:
4. Mean value theorem: if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c$ with:
5. $\frac{d}{dx}[\ln x]$ equals (for $x > 0$):