Worked example 1
Find the original function
Given dy/dx = 6x − 4 and y = 5 when x = 2, find y
- Integrate: y = 3x² − 4x + C
- Substitute x = 2 and y = 5
- Solve 5 = 12 − 8 + C.
Answer: C = 1, so y = 3x² − 4x + 1
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Higher Mathematics
Reverse derivatives with conditions
Find the original function from a derivative and a point or condition.
Before you start
- Integrate powers confidently.
- Substitute a coordinate into a function.
- Remember that + C is found from the condition.
Explanation
Sometimes you are given a derivative and a point on the original curve. Integrate first to create the family of possible functions, then use the condition to find the exact constant.
This is where + C becomes useful rather than optional.
Method and rules
- If dy/dx is known, integrate to find y.
- After integrating, substitute the given point (x, y).
- Solve for C, then write the full function.
Worked examples
Worked example 2
Check the result
Check y = 3x² − 4x + 1 for the condition
- Differentiate to get 6x − 4.
- Substitute x = 2 into y
So: The derivative matches and y(2) = 5.
Watch out
- Forgetting + C before using the condition.
- Substituting the point into dy/dx instead of y
- Finding C but not writing the final function.
- Using x and y values the wrong way round.
Exam reminder
A condition such as 'passes through (2, 5)' belongs in the integrated function, not in the derivative.