What Is a Derivative?
A derivative measures how a function changes as its input changes — it's the instantaneous rate of change. Geometrically, the derivative of a function at a point equals the slope of the tangent line to the curve at that point.
If you've ever asked "how fast is this changing right now?", you were thinking about derivatives. Speed is the derivative of position; acceleration is the derivative of speed.
The Formal Definition
The derivative of f(x) is defined as:
f'(x) = lim(h→0) [f(x + h) − f(x)] / h
This is called the limit definition or the difference quotient. While essential for understanding, in practice we use differentiation rules to find derivatives far more efficiently.
Essential Differentiation Rules
| Rule | Formula | Example |
|---|---|---|
| Power Rule | d/dx [xⁿ] = nxⁿ⁻¹ | d/dx [x³] = 3x² |
| Constant Rule | d/dx [c] = 0 | d/dx [7] = 0 |
| Sum Rule | d/dx [f + g] = f' + g' | d/dx [x² + x] = 2x + 1 |
| Product Rule | d/dx [fg] = f'g + fg' | d/dx [x·sin x] = sin x + x·cos x |
| Chain Rule | d/dx [f(g(x))] = f'(g(x))·g'(x) | d/dx [sin(x²)] = cos(x²)·2x |
Step-by-Step Example
Find the derivative of f(x) = 3x⁴ − 2x² + 5x − 8
- Apply the power rule to each term:
- d/dx [3x⁴] = 12x³
- d/dx [−2x²] = −4x
- d/dx [5x] = 5
- d/dx [−8] = 0
- f'(x) = 12x³ − 4x + 5
What the Derivative Tells You
Slope and Behavior
- If f'(x) > 0 on an interval, the function is increasing there.
- If f'(x) < 0 on an interval, the function is decreasing there.
- If f'(x) = 0 at a point, that point is a potential maximum or minimum (critical point).
Finding Maximum and Minimum Values
- Set f'(x) = 0 and solve for x (find critical points).
- Use the second derivative f''(x): if f''(x) > 0 at a critical point, it's a minimum; if f''(x) < 0, it's a maximum.
Real-World Applications
- Physics: Velocity = derivative of position; acceleration = derivative of velocity.
- Economics: Marginal cost and marginal revenue are derivatives of cost and revenue functions.
- Engineering: Optimizing design by finding where performance functions reach their peak or minimum.
- Medicine: Modeling how drug concentration in the bloodstream changes over time.
Common Derivative Values to Memorize
- d/dx [sin x] = cos x
- d/dx [cos x] = −sin x
- d/dx [eˣ] = eˣ
- d/dx [ln x] = 1/x
- d/dx [tan x] = sec²x
Once you internalize differentiation rules and develop intuition for what derivatives represent, calculus becomes less intimidating and more like a natural language for describing change.