Algebra • Topic 29

Derivatives

Derivatives measure the instantaneous rate of change. In Olympiads, they are primarily tools for optimization (finding maxima/minima) and analyzing the shape (convexity) of functions.

Basic Rules

Power Rule: $\frac{d}{d x} x^{n} = n x^{n - 1}$
Product Rule: $(uv)^{'} = u^{'} v + u v^{'}$
Chain Rule: $\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)$

Applications

1. Optimization

To find global extrema of

f (x)

[a, b]

Find critical points where $f^{'} (x) = 0$ .
Evaluate $f$ at critical points and endpoints $a, b$ .
Compare values.

Example (Minimize)

Minimize

f (x) = x + \frac{1}{x}

for

x > 0

Proof. $f^{'} (x) = 1 - \frac{1}{x ^{2}}$ . Set $f^{'} (x) = 0 ⟹ x^{2} = 1 ⟹ x = 1$ (since $x > 0$ ). $f (1) = 2$ . As $x \to 0$ or $x \to \infty$ , $f (x) \to \infty$ . Minimum is 2. ∎

2. Concavity and Convexity

Convex (Concave Up): $f^{''} (x) \geq 0$ .

The graph lies above* its tangent lines.

Concave (Concave Down): $f^{''} (x) \leq 0$ .

The graph lies below* its tangent lines.

This is the basis for Jensen's Inequality.

3. Mean Value Theorem (MVT)

f

is continuous on

[a, b]

and differentiable on

(a, b)

, there exists

c \in (a, b)

such that:

f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}

Practice Problems

Exercise (Problem 1)

Find the maximum value of

f (x) = x^{2} (1 - x)

for

x \in [0, 1]

Exercise (Problem 2)

Use derivatives to prove that

e^{x} \geq 1 + x

for all real

x

Exercise (Problem 3)

Let

P (x)

be a polynomial with real roots. Prove that

P^{'} (x)

also has only real roots. (Rolle's Theorem).