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
Higher Mathematics
Higher Maths Reverse derivatives with conditions
Practise reverse derivatives with conditions for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Recover a function from its derivative and use an initial condition to find the constant.
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.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.