Fix the mistake: Differentiation ① (Definition of the Derivative and Basic Formulas)
Differentiation ① Definition of the Derivative at a Point and the Derivative Function
The derivative at a point f'(a) of the function y = f(x) at x = a is defined as the limit of the average rate of change.
f'(a) = lim_{h→0} (f(a+h) − f(a))/h
Considering f'(x) for every x gives the derivative function f'(x).
Example ① Find the derivative function of f(x)=x² using the definition.
Solution:(f(x+h)−f(x))/h = ((x+h)²−x²)/h = (2xh+h²)/h = 2x+h
As h→0, → f'(x) = 2x
Solution:(f(x+h)−f(x))/h = ((x+h)²−x²)/h = (2xh+h²)/h = 2x+h
As h→0, → f'(x) = 2x
Differentiability and Continuity
Differentiable at x=a ⇒ continuous at x=a. The converse does not hold（for example, y=|x| is continuous at 0 but not differentiable）.
Key Points
- Derivative at a point = slope of the tangent line
- Remember the definition: lim_{h→0} (f(a+h)−f(a))/h
- Continuity does not necessarily imply differentiability（pay attention to corners）
Practice Problems
- Find f'(x) for f(x) = 3x² using the definition.
- Find f'(2) for f(x) = 1/x using the definition.
Answers and Explanations
- Answer:6x
(3(x+h)²−3x²)/h = 6x+3h → 6x。 - Answer:−1/4
(1/(2+h)−1/2)/h = −1/(2(2+h)) → −1/4。
Previous and next lessons in this course
- Previous lesson: Fix the mistake: Limits of Functions
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