A derivative sounds intimidating — but at its heart, it answers a beautifully simple question:
"How fast is this thing changing right now?"
Speed is a derivative of position. Acceleration is a derivative of speed. The temperature rising or falling at a given moment — that's a derivative too.
What is a Derivative?
The derivative of a function f(x) at a point x is the instantaneous rate of change of f at that point. Geometrically, it's the slope of the tangent line to the curve at that point.
We write it as:
f'(x) or dy/dx or d/dx[f(x)]
The Formal Definition
The derivative is defined as a limit:
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Think of it as: "What is the slope of the line between two points as those points get infinitely close together?"
The Essential Derivative Rules
You don't need to use the limit definition every time — these rules let you differentiate any function quickly.
1. The Power Rule
For any constant n:
d/dx [xⁿ] = n · xⁿ⁻¹
Examples:
- d/dx [x³] = 3x²
- d/dx [x] = 1
- d/dx [x⁻²] = -2x⁻³
- d/dx [√x] = d/dx [x^(1/2)] = (1/2)x^(-1/2)
2. The Constant Rule
The derivative of a constant is zero.
d/dx [c] = 0
Makes sense: a constant never changes, so its rate of change is 0.
3. The Sum / Difference Rule
Differentiate term by term:
d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
Example: d/dx [x³ - 5x + 2] = 3x² - 5
4. The Product Rule
When two functions are multiplied together:
d/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)
Example: Find d/dx [x²·sin(x)]
- f(x) = x², f'(x) = 2x
- g(x) = sin(x), g'(x) = cos(x)
- Result: 2x·sin(x) + x²·cos(x)
5. The Quotient Rule
When one function divides another:
d/dx [f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]²
Memory trick: "Lo d-Hi minus Hi d-Lo, over Lo-Lo"
6. The Chain Rule
The most important rule for composite functions:
d/dx [f(g(x))] = f'(g(x)) · g'(x)
Example: Find d/dx [sin(x²)]
- Outer function: sin(u), derivative: cos(u)
- Inner function: u = x², derivative: 2x
- Chain: cos(x²) · 2x = 2x·cos(x²)
Common Derivatives to Memorize
| Function | Derivative |
|---|---|
| xⁿ | n·xⁿ⁻¹ |
| eˣ | eˣ |
| ln(x) | 1/x |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec²(x) |
| aˣ | aˣ·ln(a) |
Real-World Applications
Physics: If position is s(t), then velocity is s'(t) and acceleration is s''(t).
Economics: If profit is P(q), then marginal profit is P'(q) — how much more you earn per additional unit sold.
Engineering: Derivatives are used to find the minimum and maximum of functions (e.g., minimizing material cost or maximizing structural strength).
Machine Learning: Gradient descent — the algorithm that trains neural networks — is entirely built on derivatives.
Tips for Success
Always identify the rule first — before calculating, determine if you need Power, Product, Quotient, or Chain rule.
Chain Rule is everywhere — any time you see a function inside a function, Chain Rule applies.
Don't forget constants — d/dx[5x²] = 5·2x = 10x. The constant multiplies through.
Check with the definition — if unsure about a result, verify it using the limit definition for a specific value.
Use our Derivative Rules Cheatsheet for a full printed reference, or try the Derivative Calculator to check your work step by step.