This topic covers the basic definition of derivatives, types of derivatives (futures, forwards, options, swaps), and their role in financial markets.
Introduction to Derivatives – Study Notes
1. What Is a Derivative?
- Definition: A derivative measures how a function’s output changes as its input changes.
- Geometric interpretation: Slope of the tangent line to the graph of a function at a given point.
- Notation:
- (f'(x)) or (\dfrac{df}{dx}) – derivative of (f) with respect to (x)
- (\displaystyle \frac{d}{dx}\bigl[f(x)\bigr]) – operator form
2. Fundamental Concepts
| Concept |
Description |
Symbol |
| Limit definition |
(\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}) |
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| Continuity |
A function must be continuous at (x) for the derivative to exist there. |
— |
| Differentiability |
Existence of a finite limit above; implies local linearity. |
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| Tangent line |
Line that just “touches” the curve; equation: (y = f(a) + f'(a)(x-a)). |
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3. Basic Differentiation Rules
| Rule |
Formula |
Example |
| Constant rule |
(\dfrac{d}{dx}[c] = 0) |
(\dfrac{d}{dx}[5]=0) |
| Power rule |
(\dfrac{d}{dx}[x^n] = n x^{,n-1}) |
(\dfrac{d}{dx}[x^3]=3x^2) |
| Constant multiple |
(\dfrac{d}{dx}[c,f(x)] = c,f'(x)) |
(\dfrac{d}{dx}[4\sin x]=4\cos x) |
| Sum/Difference |
(\dfrac{d}{dx}[f\pm g] = f' \pm g') |
(\dfrac{d}{dx}[x^2+\ln x]=2x+\frac1x) |
| Product rule |
((fg)' = f'g + fg') |
(\dfrac{d}{dx}[x\sin x]=\sin x + x\cos x) |
| Quotient rule |
(\displaystyle \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}) |
(\dfrac{d}{dx}!\left[\frac{x}{e^x}\right]=\frac{e^x - x e^x}{e^{2x}} = \frac{1-x}{e^x}) |
| Chain rule |
(\dfrac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)) |
(\dfrac{d}{dx}[\sin(x^2)] = \cos(x^2)\cdot 2x) |
4. Common Derivatives
| Function |
Derivative |
| ( \sin x) |
( \cos x) |
| ( \cos x) |
(-\sin x) |
| ( \tan x) |
(\sec^2 x) |
| ( e^x) |
( e^ |