# Differentiation and Integration of Powers of x

> Edexcel International GCSE Further Pure Mathematics · 4PM1
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s9-differentiation-and-integration-of-powers/

This guide teaches you core differentiation and integration rules for Edexcel IGCSE FPM 4PM1, including power rules for xⁿ, trigonometric sin(kx)/cos(kx) rules, exponential e^(kx) differentiation, definite integrals and constants of integration.

**Prerequisites:** [Fluent with indices rules for negative and fractional exponents](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s1-indices/); [Familiar with radians for trigonometric calculations](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s4-trigonometry-radians/)

## Learning objectives

- Differentiate sums of multiples of xⁿ (including negative and fractional powers), sin(kx), cos(kx) and e^(kx)
- Integrate sums of multiples of xⁿ (n≠-1), sin(kx) and cos(kx), including finding the constant of integration
- Evaluate definite integrals correctly using standard calculus rules for 4PM1
- Avoid common exam pitfalls with notation, sign errors and unit mismatches for trigonometric calculus

## Differentiation of Powers of x

**Power Rule for Differentiation** — The derivative of a term \(ax^n\) with respect to x is the product of the coefficient a, the power n, and x raised to the power \(n-1\). Applies to all integer and fractional values of n, positive or negative.

*Notation:* \frac{d}{dx}(ax^n) = anx^{n-1}

*Example:* \(\frac{d}{dx}(3x^4) = 12x^3\)

You can apply this rule to every term in a polynomial or sum of terms individually, then combine the results. Remember to rewrite any terms with roots or denominators as fractional or negative powers first to apply the rule correctly.

> **tip**
>
> Always rewrite \(\sqrt{x}\) as \(x^{1/2}\) and \(\frac{1}{x^3}\) as \(x^{-3}\) before differentiating to avoid mistakes with exponent arithmetic.

**Worked example:** Differentiate \(y = 5x^3 + 2\sqrt{x} - \frac{4}{x^2}\) with respect to x.

1. Rewrite all terms in standard \(ax^n\) form:
2. $$y = 5x^3 + 2x^{1/2} - 4x^{-2}$$
3. Apply the power rule to each term individually:
4. $$\frac{dy}{dx} = (5\times 3)x^{3-1} + (2\times \frac{1}{2})x^{\frac{1}{2}-1} - (4\times (-2))x^{-2-1}$$
5. Simplify each coefficient and exponent:
6. $$\frac{dy}{dx} = 15x^2 + x^{-1/2} + 8x^{-3}$$
7. Rewrite back to fractional/denominator form if required:
8. $$\frac{dy}{dx} = 15x^2 + \frac{1}{\sqrt{x}} + \frac{8}{x^3}$$

> **Exam tip:** You do not need to simplify answers unless explicitly asked, but simplified terms make checking easier and reduce the risk of error in follow-up questions.

*Calculator:* allowed

## Differentiation of Trigonometric and Exponential Functions

**Trigonometric and Exponential Differentiation Rules** — For constant k, the derivative rules are: \(\frac{d}{dx}(\sin kx) = k\cos kx\), \(\frac{d}{dx}(\cos kx) = -k\sin kx\), \(\frac{d}{dx}(e^{kx}) = ke^{kx}\). All angles are in radians for these rules.

*Example:* \(\frac{d}{dx}(\cos 3x) = -3\sin 3x\)

These rules are applied the same way as the power rule: differentiate each term in a sum individually, multiplying by any constant coefficient attached to the trigonometric or exponential term.

> **warning**
>
> Remember the negative sign when differentiating cos(kx): it is the most frequently missed detail in trigonometric differentiation questions for this topic.

**Worked example:** Find \(\frac{dy}{dx}\) for \(y = 2\sin 4x - 5e^{2x} + 7\cos \frac{x}{2}\)

1. Apply the differentiation rule to each term separately:
2. $$\frac{dy}{dx} = 2\times 4\cos 4x - 5\times 2 e^{2x} + 7\times (-\frac{1}{2}\sin \frac{x}{2})$$
3. Simplify coefficients:
4. $$\frac{dy}{dx} = 8\cos 4x -10e^{2x} - \frac{7}{2}\sin \frac{x}{2}$$

> **Exam tip:** You do not need to prove these rules for the 4PM1 exam, just recall them accurately as they are not provided on the formula sheet.

*Calculator:* allowed

## Indefinite Integration of Powers and Trig Functions

**Integration Rules** — 1. Power rule for integration: \(\int ax^n dx = \frac{a x^{n+1}}{n+1} + c\) for \(n \neq -1\). 2. Trigonometric integration rules: \(\int \sin kx dx = -\frac{1}{k}\cos kx + c\), \(\int \cos kx dx = \frac{1}{k}\sin kx + c\). The constant of integration +c is mandatory for all indefinite integrals.

*Example:* \(\int 6x^2 dx = 2x^3 + c\)

Integration is the reverse process of differentiation, so you can check your answer by differentiating it to see if you get the original integrand. Remember that integrating \(x^n\) increases the power by 1, unlike differentiation which decreases it. Do NOT integrate \(e^{kx}\) or \(1/x\) for this exam, as these are out of scope for 4PM1.

> **mnemonic**
>
> Power rule integration mnemonic: **I**ncrease the power, **D**ivide by the new power, **A**dd +c = IDA.

**Worked example:** Find the indefinite integral \(\int (3x^5 + 2\sin 3x - \frac{5}{x^4}) dx\)

1. Rewrite all terms to standard form for integration:
2. $$\int (3x^5 + 2\sin 3x - 5x^{-4}) dx$$
3. Apply the relevant integration rule to each term:
4. $$= \frac{3x^{5+1}}{5+1} + 2\times(-\frac{1}{3}\cos 3x) - \frac{5x^{-4+1}}{-4+1} + c$$
5. Simplify each term:
6. $$= \frac{3x^6}{6} - \frac{2}{3}\cos 3x - \frac{5x^{-3}}{-3} + c$$
7. $$= \frac{1}{2}x^6 - \frac{2}{3}\cos 3x + \frac{5}{3x^3} + c$$
8. Check by differentiating the result to confirm it matches the original integrand.

> **Exam tip:** Always include +c for all indefinite integrals, even if you forget everything else: it is an easy mark to lose if omitted.

*Calculator:* allowed

## Definite Integrals

**Definite Integral** — A definite integral calculates the signed area between the curve f(x) and the x-axis between limits x=a and x=b, where F(x) is the antiderivative of f(x). No constant of integration is needed, as it cancels out when subtracting F(a) from F(b).

*Notation:* \int_a^b f(x) dx = [F(x)]_a^b = F(b) - F(a)

*Example:* \(\int_1^2 2x dx = [x^2]_1^2 = 2^2 - 1^2 = 3\)

To evaluate a definite integral, first find the antiderivative of the integrand (as you would for an indefinite integral, without writing +c), substitute the upper limit b, then subtract the value of the antiderivative at the lower limit a. For trigonometric integrals, ensure your calculator is set to radians when evaluating the result.

**Worked example:** Evaluate \(\int_0^{\pi/2} (4x^3 + 2\cos 2x) dx\)

1. Find the antiderivative of each term first:
2. $$F(x) = \frac{4x^4}{4} + 2\times \frac{1}{2}\sin 2x = x^4 + \sin 2x$$
3. Apply the limits, upper limit minus lower limit:
4. $$[x^4 + \sin 2x]_0^{\pi/2} = \left((\frac{\pi}{2})^4 + \sin(2\times\frac{\pi}{2})\right) - \left(0^4 + \sin(0)\right)$$
5. Simplify using trigonometric values (radians):
6. $$= \frac{\pi^4}{16} + \sin(\pi) - 0 - 0 = \frac{\pi^4}{16} \approx 6.088 \text{ (3 d.p.)}$$

> **Exam tip:** If the question asks for an exact value, leave your answer in terms of π or trigonometric constants, do not convert to a decimal unless explicitly instructed.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Omitting +c for indefinite integrals
  - Why it fails: The mark scheme awards 1 mark specifically for +c on all indefinite integral questions, so omitting it costs you an easy mark
  - Correct: Add +c as the final step of every indefinite integral calculation, even if you are unsure about the rest of your answer.
- **Wrong:** Forgetting the negative sign when differentiating cos(kx) or integrating sin(kx)
  - Why it fails: Both rules include a negative sign that is easy to miss, leading to sign errors in your final answer
  - Correct: Double-check trigonometric differentiation and integration answers by reversing the operation to verify the sign is correct.
- **Wrong:** Using degrees instead of radians for trigonometric calculus questions
  - Why it fails: All calculus rules for sin(kx) and cos(kx) are defined for radians, so using degrees gives incorrect numerical results
  - Correct: Set your calculator to radians before starting any calculus question involving trigonometric functions.
- **Wrong:** Decreasing the power of x when integrating instead of increasing it
  - Why it fails: Confusing integration and differentiation power rules leads to incorrect exponents
  - Correct: Use the IDA mnemonic: Increase the power, Divide by the new power, Add +c for integration.
- **Wrong:** Subtracting the upper limit from the lower limit when calculating definite integrals
  - Why it fails: The definite integral is defined as upper limit minus lower limit, so reversing this gives the negative of the correct answer
  - Correct: Always write [F(x)]_a^b = F(b) - F(a) explicitly in your working to avoid mixing up the order of limits.

## Cheatsheet

| Operation | Rule | Key Notes |
| --- | --- | --- |
| Differentiate \(ax^n\) | \(anx^{n-1}\) | All n positive/negative/fractional |
| Differentiate \(\sin kx\) | \(k\cos kx\) | Radians only, no negative sign |
| Differentiate \(\cos kx\) | \(-k\sin kx\) | Radians only, remember negative sign |
| Differentiate \(e^{kx}\) | \(ke^{kx}\) | Not tested for integration in 4PM1 |
| Integrate \(ax^n\) (n≠-1) | \(\frac{ax^{n+1}}{n+1} + c\) | Increase power by 1, divide by new power |
| Integrate \(\sin kx\) | \(-\frac{1}{k}\cos kx + c\) | Radians only, remember negative sign |
| Integrate \(\cos kx\) | \(\frac{1}{k}\sin kx + c\) | Radians only, no negative sign |
| Definite integral \(\int_a^b f(x)dx\) | \(F(b)-F(a)\) | No +c needed, F(x) is antiderivative |

## What's next

Now that you have mastered the core differentiation and integration rules for Edexcel IGCSE Further Pure Maths 4PM1, you are ready to move on to more advanced calculus techniques and their real-world applications. The next sub-topic covers the product, quotient and chain rules for differentiation, which are required for more complex composite and combined functions, followed by applications of calculus to kinematics, area under curves and volumes of revolution. Make sure you practice a mix of differentiation and integration questions regularly to avoid mixing up the rules, and always check your working using reverse operations where possible.

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