Differentiation is the operation of finding the instantaneous rate of change. It is defined as the limit of the average rate of change.
Average Rate of Change
For y = f(x), when x changes from a to b, the average rate of change is:
(f(b) − f(a)) / (b − a)
Alternatively, using x = a and the change h: (f(a+h) − f(a)) / h
Definition of the Derivative
The derivative is obtained by taking the limit as h → 0:
f'(a) = lim_{h→0} (f(a+h) − f(a)) / h
Example ①
Find the derivative of f(x) = x² at x = 2 from the definition.
Answer:f'(2) = lim_{h→0} ((2+h)² − 4)/h = lim_{h→0} (4+4h+h² − 4)/h
= lim_{h→0} (4h + h²)/h = lim_{h→0} (4 + h) = 4
Find the derivative of f(x) = x² at x = 2 from the definition.
Answer:f'(2) = lim_{h→0} ((2+h)² − 4)/h = lim_{h→0} (4+4h+h² − 4)/h
= lim_{h→0} (4h + h²)/h = lim_{h→0} (4 + h) = 4
Example ②
Find the derivative f'(x) of f(x) = x³ from the definition.
Answer:f'(x) = lim_{h→0} ((x+h)³ − x³)/h
= lim_{h→0} (3x²h + 3xh² + h³)/h = lim_{h→0} (3x² + 3xh + h²) = 3x²
Find the derivative f'(x) of f(x) = x³ from the definition.
Answer:f'(x) = lim_{h→0} ((x+h)³ − x³)/h
= lim_{h→0} (3x²h + 3xh² + h³)/h = lim_{h→0} (3x² + 3xh + h²) = 3x²
Key Points
- Average rate of change: Δy/Δx = (f(a+h)−f(a))/h
- Derivative: the limit as h→0 (the instantaneous rate of change)
- The derivative is the “slope function,” giving the slope of the tangent line
Practice Problems
- Find the derivative of f(x) = 2x² at x = 3 from the definition.
- Find the derivative of f(x) = x² + x from the definition.
- Find the derivative of f(x) = 1/x from the definition.
Answers and Explanations
- Answer:12
Explanation: lim(2(3+h)²−18)/h = lim(2(9+6h+h²)−18)/h = lim(12h+2h²)/h = 12。 - Answer:2x + 1
Explanation: lim((x+h)²+(x+h)−x²−x)/h = lim(2xh+h²+h)/h = 2x+1。 - Answer:−1/x²
Explanation: lim(1/(x+h)−1/x)/h = lim((x−(x+h))/(x(x+h)))/h = lim(−1/(x(x+h))) = −1/x²。