# Differentiation (Edexcel IAL Pure Mathematics 3)

> Edexcel International A-Level Mathematics · IAL P3 Maths 2018 Spec
> Source: https://www.owlsprep.com/study/edexcel-ial-math-p3-differentiation/

This guide covers all Edexcel IAL P3 differentiation content, including standard function derivatives, chain/product/quotient rules, inverse differentiation, and exponential growth/decay models for the 2018 WMA13 syllabus.

**Prerequisites:** [Basic polynomial differentiation (P1)](https://www.owlsprep.com/study/edexcel-ial-math-p1-basic-differentiation/); [Stationary points and applications (P2)](https://www.owlsprep.com/study/edexcel-ial-math-p2-stationary-points/)

## Learning objectives

- Differentiate standard P3 functions including $e^{kx}$, $\ln  kx$, trigonometric functions and their reciprocals
- Apply chain, product and quotient rules to composite, product and quotient functions
- Use the $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$ identity for inverse differentiation
- Solve exponential growth/decay problems, analyze model appropriateness and apply $\frac{d}{dx}(a^x) = a^x \ln a$ rule

## Differentiation of Standard P3 Functions

Start by memorizing the standard derivatives you are required to know (not provided in the formula booklet), alongside the given derivatives for reciprocal trigonometric functions. You can differentiate any sum or difference of these functions directly using the sum rule: differentiate each term separately.

**Standard P3 Derivatives** — Required to memorize: $\frac{d}{dx}(\sin kx) = k\cos kx$, $\frac{d}{dx}(\cos kx) = -k\sin kx$, $\frac{d}{dx}(e^{kx}) = ke^{kx}$, $\frac{d}{dx}(\ln kx) = 1/x$, $\frac{d}{dx}(a^x) = a^x \ln a$. Given in formula booklet: $\frac{d}{dx}(\tan kx) = k\sec^2 kx$, $\frac{d}{dx}(\sec x) = \sec x \tan x$, $\frac{d}{dx}(\cot x) = -\cosec^2 x$, $\frac{d}{dx}(\cosec x) = -\cosec x \cot x$.

**Worked example:** Differentiate $f(x) = 3e^{2x} + 4\cos 3x + \ln 5x$ with respect to $x$.

1. Differentiate each term separately using the sum rule:
2. $$\frac{d}{dx}(3e^{2x}) = 3 \times 2e^{2x} = 6e^{2x}$$
3. $$\frac{d}{dx}(4\cos 3x) = 4 \times (-3\sin 3x) = -12\sin 3x$$
4. $$\frac{d}{dx}(\ln 5x) = \frac{1}{x}$$
5. Combine terms for the final derivative:
6. $$f'(x) = 6e^{2x} - 12\sin 3x + \frac{1}{x}$$

> **Exam tip:** Remember the derivative of $\ln  kx$ is $1/x$, not $k/x$, as the $k$ cancels out when applying the chain rule or splitting $\ln  k + \ln  x$.

*Calculator:* allowed

## Applying Chain, Product and Quotient Rules

For composite functions, products of two functions, or quotients of two functions, use the corresponding differentiation rule. You may need to combine multiple rules for more complex functions, as is common in exam questions.

**Worked example:** Differentiate $y = \tan^2(2x)$ with respect to $x$.

1. Rewrite as a composite function: $y = [\tan(2x)]^2$, let $u = \tan(2x)$, $y = u^2$
2. Apply chain rule: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$
3. $$\frac{dy}{du} = 2u = 2\tan(2x)$$
4. (using the given derivative of $\tan kx$ with $k=2$)

   $$\frac{du}{dx} = 2\sec^2(2x)$$
5. Multiply results and substitute back $u$:
6. $$\frac{dy}{dx} = 2\tan(2x) \times 2\sec^2(2x) = 4\tan(2x)\sec^2(2x)$$

**Worked example:** Differentiate $y = 2x^4 \sin x$ with respect to $x$.

1. Apply product rule: let $u = 2x^4$, $v = \sin x$, so $\frac{dy}{dx} = u'v + uv'$
2. $$u' = 8x^3, v' = \cos x$$
3. $$\frac{dy}{dx} = 8x^3\sin x + 2x^4\cos x = 2x^3(4\sin x + x\cos x)$$

> **Exam tip:** Label functions clearly (e.g. $u =$ first term, $v =$ second term) for product/quotient rule questions to avoid mixing up terms in your working.

*Calculator:* allowed

## Inverse Differentiation: $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$

When given $x$ as a function of $y$, you can find $\frac{dy}{dx}$ by first calculating $\frac{dx}{dy}$ then taking its reciprocal. This avoids rearranging to get $y$ as a function of $x$, which is often impossible for trigonometric or exponential functions.

**Inverse Differentiation Rule** — For a continuous function where $\frac{dx}{dy} ≠ 0$, $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$. Final answers should be in terms of $x$ unless specified otherwise.

**Worked example:** Find $\frac{dy}{dx}$ for $x = \sin 3y$, giving your answer in terms of $x$.

1. First differentiate $x$ with respect to $y$:
2. $$\frac{dx}{dy} = 3\cos 3y$$
3. Take reciprocal to get $\frac{dy}{dx}$ in terms of $y$:
4. $$\frac{dy}{dx} = \frac{1}{3\cos 3y}$$
5. Use identity $\sin^2θ + \cos^2θ = 1$ to rewrite $\cos 3y$ in terms of $x$: $\cos 3y = \sqrt{1 - \sin^2 3y} = \sqrt{1 - x^2}$
6. Substitute back for final answer in terms of $x$:
7. $$\frac{dy}{dx} = \frac{1}{3\sqrt{1 - x^2}}$$

> **Exam tip:** Always check if the question requires your answer in terms of $x$; if not, leaving it in terms of $y$ is acceptable, but conversion to $x$ is usually expected for full marks.

*Calculator:* allowed

## Exponential Growth and Decay Models

Exponential models are given in the form $N = N_0e^{kt}$, where $N_0$ is the initial value at $t=0$, $k$ is positive for growth and negative for decay. You will need to differentiate these models, analyze their behavior for large $t$, and evaluate how appropriate the model is for real-world use.

> **info**
>
> The derivative of $a^x$ is $a^x \ln a$, which is used for exponential functions with a base other than $e$ that may appear in model questions.

**Worked example:** The number of bacteria in a sample is modeled by $N = 200e^{0.05t}$, where $t$ is time in hours. (a) Find the initial number of bacteria. (b) Find the rate of change of $N$ with respect to $t$ when $t=10$, giving your answer to 3 significant figures. (c) Explain why this model is not appropriate for very large values of $t$.

1. (a) Initial value is at $t=0$:
2. $$N_0 = 200e^0 = 200$$
3. (b) Differentiate $N$ to get the rate of change:
4. $$\frac{dN}{dt} = 200 \times 0.05 e^{0.05t} = 10e^{0.05t}$$
5. Substitute $t=10$:
6. ($\approx 16.5$ bacteria per hour, 3 s.f.)

   $$\frac{dN}{dt} = 10e^{0.5} \approx 16.5$$
7. (c) For very large $t$, $N$ grows without bound, which is impossible as bacteria will run out of space and nutrients, so the model overestimates population for large $t$.

> **Exam tip:** When evaluating model appropriateness, always refer to real-world constraints: population limits, non-negative quantities, or physical bounds the exponential model does not account for.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting the negative sign when differentiating $\cos kx$, $\cot x$ or $\cosec x$
  - Why it fails: Mixing up negative derivatives of cosine, cotangent, cosecant with positive derivatives of sine, tangent, secant
  - Correct: Write the sign first when differentiating these functions, then add the rest of the derivative
- **Wrong:** Calculating $\frac{d}{dx}(\ln kx) = \frac{k}{x}$
  - Why it fails: Misapplying the chain rule, forgetting $\ln  kx = \ln  k + \ln  x$ so the derivative of constant $\ln  k$ is 0
  - Correct: Memorize $\frac{d}{dx}(\ln kx) = \frac{1}{x}$ regardless of $k$, or apply chain rule correctly: $\frac{1}{kx} \times k = \frac{1}{x}$
- **Wrong:** Writing quotient rule numerator as $fg' - f'g$ instead of $f'g - fg'$
  - Why it fails: Confusing the order of terms leading to sign errors
  - Correct: Use the mnemonic: "Low dHigh minus High dLow, over the square of what's below"
- **Wrong:** Leaving $\frac{dy}{dx}$ in terms of $y$ when the question asks for it in terms of $x$
  - Why it fails: Forgetting to convert $y$-dependent terms using trigonometric/algebraic identities after taking the reciprocal
  - Correct: Check question requirements explicitly, and use relevant identities to substitute $y$ with $x$ where needed
- **Wrong:** Applying power rule to $a^x$ to get $xa^{x-1}$
  - Why it fails: Confusing exponential functions (variable in exponent) with power functions (variable in base)
  - Correct: Identify if the variable is in the base (use power rule) or exponent (use $\frac{d}{dx}(a^x) = a^x \ln a$)

## Cheatsheet

| Rule/Function | Derivative | Given in Formula Booklet? |
| --- | --- | --- |
| $\sin kx$ | $k\cos kx$ | No |
| $\cos kx$ | $-k\sin kx$ | No |
| $e^{kx}$ | $ke^{kx}$ | No |
| $\ln kx$ | $\frac{1}{x}$ | No |
| $a^x$ | $a^x \ln a$ | No |
| $\tan kx$ | $k\sec^2 kx$ | Yes |
| $\sec x$ | $\sec x \tan x$ | Yes |
| $\cot x$ | $-\cosec^2 x$ | Yes |
| $\cosec x$ | $-\cosec x \cot x$ | Yes |
| Product Rule ($fg$) | $f'g + fg'$ | No |
| Quotient Rule ($f/g$) | $\frac{f'g - fg'}{g^2}$ | Yes |
| Chain Rule ($f(g(x))$) | $f'(g(x))g'(x)$ | No |
| Inverse Differentiation | $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$ | No |

## What's next

Now that you have mastered P3 differentiation content, you can move on to advanced P4 differentiation topics including implicit and parametric differentiation, as well as connected rates of change. You should also practice applying these rules to past paper P3 questions to familiarize yourself with exam phrasing and mark scheme requirements. Make sure you memorize the non-given derivatives before your exam, as they are frequently tested across multiple question types. Exponential growth/decay questions often appear alongside logarithm questions, so revising P3 logarithms will help you answer full multi-part questions efficiently.

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