Differentiating the derivative function f'(x) gives the second derivative f''(x), and differentiating that gives the third derivative f'''(x), and so on.
For the point (a, f(a)) on the graph of y = f(x), the equation of the tangent line is
y − f(a) = f'(a)·(x − a)
Example ① Find the equation of the tangent line to f(x) = x³ − 2x at x = 1.
Solution:f(1) = −1、f'(x) = 3x²−2、f'(1) = 1。
Tangent line：y − (−1) = 1·(x − 1) → y = x − 2
Solution:f(1) = −1、f'(x) = 3x²−2、f'(1) = 1。
Tangent line：y − (−1) = 1·(x − 1) → y = x − 2
Example ② Find f''(x) and f'''(x) for f(x) = x⁴.
Solution:f'(x) = 4x³、f''(x) = 12x²、f'''(x) = 24x
Solution:f'(x) = 4x³、f''(x) = 12x²、f'''(x) = 24x
Key Points
- Slope of the tangent line = derivative at that point f'(a)
- The second derivative f''(x) is used to determine acceleration and concavity
Practice Problems
- Find the tangent line to y = x² − 3x at x = 2
- Find f''(x) for f(x) = x⁵
Answers and Explanations
- Answer:y = x − 4
f(2)=−2、f'(x)=2x−3、f'(2)=1。y−(−2)=1·(x−2)。 - Answer:20x³