Common Mistakes
Identify and avoid the most frequent errors students make when computing derivatives. Learn strategies to catch and correct your own mistakes.
Learn
Even strong students make errors when working with derivatives. By learning to recognize common mistakes, you can check your work more effectively and avoid losing points on tests.
Categories of Common Errors
- Rule Application Errors: Using the wrong differentiation rule
- Chain Rule Omission: Forgetting to apply the chain rule
- Algebraic Mistakes: Errors in simplification
- Sign Errors: Losing negative signs
- Exponent Errors: Mistakes with fractional or negative exponents
- Constant Confusion: Treating variables as constants or vice versa
Error Prevention Strategies
- Write out each step explicitly
- Check your answer by substituting a simple value
- Verify the derivative matches the original function's behavior
- Read the problem again after solving
Examples
Mistake 1: Forgetting the Chain Rule
Problem: Find the derivative of f(x) = (3x + 1)^5
WRONG: f'(x) = 5(3x + 1)^4
Missing the derivative of the inside!
CORRECT: f'(x) = 5(3x + 1)^4 * 3 = 15(3x + 1)^4
Always multiply by the derivative of the inner function
Mistake 2: Product Rule Confusion
Problem: Find the derivative of f(x) = x^2 * sin(x)
WRONG: f'(x) = 2x * cos(x)
This is not how product rule works!
CORRECT: f'(x) = 2x * sin(x) + x^2 * cos(x)
Product rule: (fg)' = f'g + fg'
Mistake 3: Negative Exponent Errors
Problem: Find the derivative of f(x) = 1/x^3
WRONG: f'(x) = 1/(3x^2) or f'(x) = -3x^2
Exponent rules applied incorrectly
CORRECT: f(x) = x^(-3), so f'(x) = -3x^(-4) = -3/x^4
Rewrite as a negative exponent first, then apply power rule
Mistake 4: Square Root Errors
Problem: Find the derivative of f(x) = sqrt(x)
WRONG: f'(x) = sqrt(1) = 1 or f'(x) = 1/(2*sqrt(x))*0
Not recognizing sqrt as a power
CORRECT: f(x) = x^(1/2), so f'(x) = (1/2)x^(-1/2) = 1/(2*sqrt(x))
sqrt(x) = x^(1/2), then apply power rule
Mistake 5: Quotient Rule Sign Error
Problem: Find the derivative of f(x) = x/(x+1)
WRONG: f'(x) = [1*(x+1) + x*1]/(x+1)^2 = (2x+1)/(x+1)^2
Used + instead of - in quotient rule
CORRECT: f'(x) = [1*(x+1) - x*1]/(x+1)^2 = 1/(x+1)^2
Quotient rule: (f/g)' = (f'g - fg')/g^2 (minus in the middle!)
Practice
Each problem below contains a common mistake. Find the error and correct it.
Problem 1: Find the Error
Student work for f(x) = (2x - 1)^4:
"f'(x) = 4(2x - 1)^3"
Show Error and Correction
Error: Forgot chain rule - missing derivative of inner function (2x - 1)
Correct: f'(x) = 4(2x - 1)^3 * 2 = 8(2x - 1)^3
Problem 2: Find the Error
Student work for f(x) = 3x^4 - 2x^3:
"f'(x) = 12x^4 - 6x^3"
Show Error and Correction
Error: Did not decrease the exponents by 1
Correct: f'(x) = 12x^3 - 6x^2
Problem 3: Find the Error
Student work for f(x) = 5:
"f'(x) = 5"
Show Error and Correction
Error: Derivative of a constant is 0, not the constant itself
Correct: f'(x) = 0
Problem 4: Find the Error
Student work for f(x) = x * e^x:
"f'(x) = 1 * e^x = e^x"
Show Error and Correction
Error: Used power rule thinking instead of product rule
Correct: f'(x) = 1*e^x + x*e^x = e^x(1 + x)
Problem 5: Find the Error
Student work for f(x) = sqrt(x^2 + 1):
"f'(x) = 1/(2*sqrt(x^2 + 1))"
Show Error and Correction
Error: Forgot chain rule for the inner function x^2 + 1
Correct: f'(x) = 1/(2*sqrt(x^2 + 1)) * 2x = x/sqrt(x^2 + 1)
Problem 6: Find the Error
Student work for f(x) = 1/(x^2):
"f'(x) = 1/(2x)"
Show Error and Correction
Error: Incorrectly applied derivative to denominator only
Correct: f(x) = x^(-2), so f'(x) = -2x^(-3) = -2/x^3
Problem 7: Find the Error
Student work for f(x) = (x+2)/(x-1):
"f'(x) = [1*(x-1) + (x+2)*1]/(x-1)^2 = (2x+1)/(x-1)^2"
Show Error and Correction
Error: Used + instead of - in quotient rule
Correct: f'(x) = [1*(x-1) - (x+2)*1]/(x-1)^2 = -3/(x-1)^2
Problem 8: Find the Error
Student work for f(x) = x^(3/2):
"f'(x) = (3/2)*x^(3/2 - 1) = (3/2)*x^(2/2) = (3/2)*x"
Show Error and Correction
Error: Subtraction of fractions: 3/2 - 1 = 3/2 - 2/2 = 1/2, not 2/2
Correct: f'(x) = (3/2)*x^(1/2) = (3/2)*sqrt(x)
Problem 9: Spot All Errors
Find the derivative of f(x) = (x^2 + 3)^2:
Student work: "Let u = x^2 + 3. Then f = u^2, so f' = 2u = 2(x^2 + 3)"
Show Errors and Correction
Error: Applied chain rule but forgot to multiply by du/dx = 2x
Correct: f'(x) = 2(x^2 + 3) * 2x = 4x(x^2 + 3)
Problem 10: Verification Practice
Verify whether this derivative is correct: If f(x) = x^3 - 3x, is f'(x) = 3x^2 - 3?
Show Verification
Method: Check at a point. At x = 2:
f(2) = 8 - 6 = 2
f'(2) = 12 - 3 = 9
Check slope: [f(2.001) - f(2)]/0.001 = [(8.012 - 6.003) - 2]/0.001 = 0.009/0.001 = 9
Conclusion: Yes, the derivative f'(x) = 3x^2 - 3 is correct
Check Your Understanding
Answer these reflection questions:
- What is the most common mistake you personally make with derivatives?
- How can you remember when to use the chain rule?
- What is the key difference between the product rule and quotient rule?
- Name two ways to verify that your derivative is correct.
Next Steps
- Create a personal error log to track your mistakes
- Before submitting any problem, check for these common errors
- Practice the types of problems where you make the most mistakes
- Proceed to the Unit Quiz to test your overall understanding