Differentiate powers and sums of powers
Apply the power rule to rational powers, constants and sums, including negative powers, while observing where a function is defined.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the derivative of a constant k?
What you will learn
- Differentiate xⁿ for any rational n in its domain
- Differentiate constant multiples and sums term by term
- Rewrite reciprocals as powers before differentiating
Bring the power down and reduce it by one
For a rational power n, d(xⁿ)/dx = nxⁿ⁻¹ wherever the expression is defined. A constant has derivative zero; a constant multiple keeps that multiplier.
For x⁻², the derivative is −2x⁻³. Rewriting 1/x² as x⁻² helps keep the power rule consistent. Retain domain restrictions when fractional powers or denominators are present.
Differentiate each term separately
Differentiation is linear: differentiate each term of a sum or difference, keeping its coefficient and sign. For 3x⁴ − 2x² + 7, the derivative is 12x³ − 4x.
A constant term disappears because it does not change as x changes. The 0606 syllabus does not provide a calculus formula sheet, so learners need to know and apply the standard derivative rules.
Pause and explain
Rewrite 1/x³ as a power of x.
Worked example
Differentiate y = 3x⁵ − 4x⁻² + 7x − 6.
Show the worked solution
- Apply the power rule to 3x⁵: its derivative is 15x⁴.
- The derivative of −4x⁻² is 8x⁻³; the derivative of 7x is 7.
- The constant −6 has derivative zero.
Answer dy/dx = 15x⁴ + 8x⁻³ + 7.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Differentiate y = 5x⁴ − 3x² + 2x − 9.
Give me a hint
Apply the power rule term by term; the constant vanishes.
Compare my reasoning
- 5x⁴ becomes 20x³.
- −3x² becomes −6x and 2x becomes 2.
- −9 becomes 0.
dy/dx = 20x³ − 6x + 2.
Look for these in your work
- Coefficients multiplied by powers
- Exponents reduced once
- Constant term removed
Differentiate y = 2x^(3/2) + 6x^(-1/2). State a domain on which the original function is real and defined.
Give me a hint
Use the power rule and remember x appears in a denominator after the second term is simplified.
Compare my reasoning
- The derivative is 3x^(1/2) − 3x^(−3/2).
- The original square root requires x ≥ 0 and the denominator requires x ≠ 0.
- Together the original function is defined for x > 0.
dy/dx = 3√x − 3/x^(3/2), for the original domain x > 0.
Look for these in your work
- Fractional powers differentiated
- Negative exponent kept
- Original domain stated
A height model is h(t) = 4t³ − 6t² + 2t metres. Find its instantaneous rate of change at t = 2 seconds.
Give me a hint
Differentiate with respect to t, then substitute t = 2.
Compare my reasoning
- h′(t) = 12t² − 12t + 2.
- h′(2) = 12(4) − 24 + 2.
- Evaluate and retain metres per second.
h′(2) = 26 m/s.
Look for these in your work
- Function differentiated correctly
- Input value substituted
- Rate units stated
Common mistakes
- Subtracting one from the coefficient instead of the exponent
- Forgetting the negative sign in a negative power
- Differentiating a constant as if it were x
- Ignoring where a fractional power is real or a denominator is nonzero
What can you explain now?
What is d(x⁷)/dx?
Compare with the explanation
7x⁶
Multiply by the exponent and reduce the exponent by one.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How does the power rule work for positive, zero and negative exponents?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.