Differentiation
Key Formula
dxd[f(x)]=h→0limhf(x+h)−f(x)
Where:
- f(x) = The function being differentiated
- x = The independent variable
- h = A small increment that approaches zero
Worked Example
Problem: Differentiate f(x) = x³ with respect to x.
Step 1: Apply the power rule: bring the exponent down as a coefficient and reduce the exponent by 1.
dxd[x3]=3x3−1
Step 2: Simplify the exponent.
3x3−1=3x2
Answer: The derivative is f'(x) = 3x². This means, for example, at x = 2 the function is changing at a rate of 3(2)² = 12 units per unit increase in x.
Why It Matters
Differentiation is central to physics, engineering, and economics because it quantifies how quantities change. For instance, differentiating a position function gives velocity, and differentiating a cost function reveals marginal cost. Nearly every optimization problem — finding maximum profit, minimum distance, or best fit — relies on setting a derivative equal to zero.
Common Mistakes
Mistake: Confusing differentiation with taking a difference. Students sometimes subtract two function values instead of applying derivative rules.
Correction: Differentiation uses specific rules (power rule, product rule, chain rule, etc.) or the limit definition to find an expression for the instantaneous rate of change, not just the change between two points.
Related Terms
- Derivative — The result produced by differentiation
- Integration — The reverse process of differentiation
- Chain Rule — A key rule used in differentiation
- Power Rule — The most basic differentiation rule
- Limit — The foundational concept behind differentiation