Differentiation (Edexcel IAL Pure Mathematics 3)

Q: Which differentiation formulas are given in the P3 exam?

The formula booklet provides derivatives of , , , , and the quotient rule. You must memorize derivatives of , , , , sum, product, chain, and rules.

Edexcel International A-Level Mathematics· WMA13 Unit P3 §4.1 to §4.4· 25 min read

1. Differentiation of Standard P3 Functions★★☆☆☆⏱ 5 min

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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.

📘 Definition

Standard P3 Derivatives

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

📐 Worked Example

Differentiate $f (x) = 3 e^{2 x} + 4 cos 3 x + ln 5 x$ 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 $ln k x$ is $1/ x$ , not $k / x$ , as the $k$ cancels out when applying the chain rule or splitting $ln k + ln x$ .

2. Applying Chain, Product and Quotient Rules★★★☆☆⏱ 7 min

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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 $y = tan^{2} (2 x)$ with respect to $x$ .

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

1
Apply product rule: let $u = 2 x^{4}$ , $v = sin x$ , so $\frac{d y}{d x} = u^{'} v + u v^{'}$
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.

3. Inverse Differentiation: $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$★★★☆☆⏱ 4 min

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

📘 Definition

Inverse Differentiation Rule

For a continuous function where $\frac{d x}{d y} \neq = 0$ , $\frac{d y}{d x} = \frac{1}{\frac{d x}{d y}}$ . Final answers should be in terms of $x$ unless specified otherwise.

📐 Worked Example

Find $\frac{d y}{d x}$ for $x = sin 3 y$ , 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 $\frac{d y}{d x}$ in terms of $y$ :
4
$\frac{dy}{dx} = \frac{1}{3\cos 3y}$
5
Use identity $sin^{2} θ + cos^{2} θ = 1$ to rewrite $cos 3 y$ in terms of $x$ : $cos 3 y = 1 - sin^{2} 3 y = 1 - x^{2}$
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.

4. Exponential Growth and Decay Models★★★★☆⏱ 6 min

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Exponential models are given in the form $N = N_{0} e^{k t}$ , 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.

📐 Worked Example

The number of bacteria in a sample is modeled by $N = 200 e^{0.05 t}$ , 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.

5. Common Pitfalls

Wrong move:

Forgetting the negative sign when differentiating $cos k x$ , $cot x$ or $cosec x$

Why:

Mixing up negative derivatives of cosine, cotangent, cosecant with positive derivatives of sine, tangent, secant

Correct move:

Write the sign first when differentiating these functions, then add the rest of the derivative

Wrong move:

Calculating $\frac{d}{d x} (ln k x) = \frac{k}{x}$

Why:

Misapplying the chain rule, forgetting $ln k x = ln k + ln x$ so the derivative of constant $ln k$ is 0

Correct move:

Memorize $\frac{d}{d x} (ln k x) = \frac{1}{x}$ regardless of $k$ , or apply chain rule correctly: $\frac{1}{k x} \times k = \frac{1}{x}$

Wrong move:

Writing quotient rule numerator as $f g^{'} - f^{'} g$ instead of $f^{'} g - f g^{'}$

Why:

Confusing the order of terms leading to sign errors

Correct move:

Use the mnemonic: "Low dHigh minus High dLow, over the square of what's below"

Wrong move:

Leaving $\frac{d y}{d x}$ in terms of $y$ when the question asks for it in terms of $x$

Why:

Forgetting to convert $y$ -dependent terms using trigonometric/algebraic identities after taking the reciprocal

Correct move:

Check question requirements explicitly, and use relevant identities to substitute $y$ with $x$ where needed

Wrong move:

Applying power rule to $a^{x}$ to get $x a^{x - 1}$

Why:

Confusing exponential functions (variable in exponent) with power functions (variable in base)

Correct move:

Identify if the variable is in the base (use power rule) or exponent (use $\frac{d}{d x} (a^{x}) = a^{x} ln a$ )

6. Quick Reference Cheatsheet

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

7. Frequently Asked

Which differentiation formulas are given in the P3 exam?

The formula booklet provides derivatives of $tan k x$ , $sec x$ , $cot x$ , $cosec x$ , and the quotient rule. You must memorize derivatives of $sin k x$ , $cos k x$ , $e^{k x}$ , $ln x$ , sum, product, chain, and $a^{x}$ rules.

Do I need to form differential equations for P3 exponential model questions?

No, forming differential equations is P4 content. For P3, you will be given the exponential model and asked to analyze or differentiate it.

Going deeper

formula_bookletEdexcel IAL Maths 2018 Spec Formula BookletContains all given P3 differentiation formulas

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.