1. Mechanics

1.1 Forces and equilibrium

1.2 Kinematics of motion in a straight line

1.2.1 Scalar and vector quantities in motion

1.2.2 Displacement-time and velocity-time graphs

1.2.3 Calculus in kinematics

1.2.4 Constant acceleration equations

1.3 Momentum

1.3.1 Linear momentum and conservation in one dimension

1.3.2 Direct impact and combined bodies

1.4 Newton’s laws of motion

1.4.1 Applying Newton’s laws to linear motion with constant mass

1.4.2 Mass, weight and motion on inclined planes

1.4.3 Connected particles and pulley problems

1.5 Energy, work and power

1.5.1 Work done by a force and energy concepts

1.5.2 Kinetic and potential energy calculations

1.5.3 Conservation of energy and mechanical systems

1.5.4 Power, force and velocity relationships

2. Pure Mathematics 1

2.1 Trigonometry

2.1.1 Exact values and inverse trigonometric functions

2.1.2 Trigonometric identities and solving equations

2.1.3 Graphs of sine, cosine, and tangent functions

2.2 Series

2.2.1 Binomial expansion for positive integer powers

2.2.2 Arithmetic and geometric progression formulas

2.2.3 Convergence and sum to infinity for geometric series

2.3 Differentiation

2.3.1 Gradient as a limit and first principles

2.3.2 Basic rules and chain rule for differentiation

2.3.3 Tangents, normals, and rates of change

2.3.4 Stationary points and curve sketching

2.4 Integration

2.4.1 Basic integration rules and finding constants

2.4.2 Evaluation of definite integrals

2.4.3 Area under curves and volume of revolution

2.5 Quadratics

2.5.1 Completing the square and vertex form

2.5.2 Discriminant and nature of roots

2.5.3 Solving quadratic equations and inequalities

2.5.4 Simultaneous equations involving quadratics

2.5.5 Equations quadratic in a function of x

2.6 Functions

2.6.1 Function terminology, domain and range

2.6.2 Composition and inverse of functions

2.6.3 Graphical relationship between function and its inverse

2.6.4 Graph transformations including translation, reflection and stretch

2.7 Coordinate geometry

2.7.1 Equation and forms of a straight line

2.7.2 Line and circle geometry, intersections and tangents

2.7.3 Intersections of graphs and solutions of equations

2.8 Circular measure

2.8.1 Radian measure and conversion from degrees

2.8.2 Arc length and sector area calculations

3. Pure Mathematics 2

3.1 Algebra

3.1.1 Modulus functions and solving modulus equations and inequalities

3.1.2 Polynomial division, factor theorem and remainder theorem

3.2 Logarithmic and exponential functions

3.2.1 Laws of logarithms and relationship with indices

3.2.2 Graphs and inverse relationship of ex and ln x

3.2.3 Solving equations involving logarithms and exponents

3.2.4 Transforming functions to linear form using logarithms

3.3 Trigonometry

3.3.1 Graphs and properties of all six trigonometric functions

3.3.2 Identities and expansions including compound and double angles

3.4 Differentiation

3.4.1 Derivatives of standard functions and composite functions

3.4.2 Product and quotient rules in differentiation

3.4.3 Parametric and implicit differentiation

3.5 Integration

3.5.1 Integration of standard exponential and trigonometric forms

3.5.2 Trigonometric identities in integration

3.5.3 Trapezium rule for numerical integration

3.6 Numerical solution of equations

3.6.1 Root approximation using graphical methods

3.6.2 Fixed-point iteration and convergence of sequences

4. Probability & Statistics 1

4.1 Representation of data

4.1.1 Statistical diagrams and data presentation

4.1.2 Measures of central tendency and variation

4.1.3 Cumulative frequency and interpretation

4.1.4 Calculation of mean and standard deviation

4.2 Permutations and combinations

4.2.1 Concepts and basic problems of selections

4.2.2 Arrangements with repetition and restrictions

4.3 Probability

4.3.1 Basic probability rules and enumeration

4.3.2 Addition and multiplication of probabilities

4.3.3 Exclusive and independent events

4.3.4 Conditional probability and tree diagrams

4.4 Discrete random variables

4.4.1 Probability distributions and expectation

4.4.2 Binomial and geometric distributions

4.4.3 Mean and variance of binomial and geometric distributions

4.5 The normal distribution

4.5.1 Properties and use of the normal distribution

4.5.2 Standardisation and probability calculations

4.5.3 Normal approximation to the binomial distribution

5. Probability & Statistics 2

5.1 The Poisson distribution

5.1.1 Probability calculations and properties of Poisson distribution

5.1.2 Poisson as a model and approximation to binomial

5.1.3 Normal approximation to Poisson distribution

5.2 Linear combinations of random variables

5.2.1 Expectation and variance of linear combinations

5.2.2 Distributions resulting from combinations of normal and Poisson variables

5.3 Continuous random variables

5.3.1 Probability density functions and properties

5.3.2 Calculating mean, variance, and percentiles

5.4 Sampling and estimation

5.4.1 Sampling concepts and randomness

5.4.2 Distribution and variance of the sample mean

5.4.3 Unbiased estimation of mean and variance

5.4.4 Confidence intervals for mean and proportion

5.5 Hypothesis tests

5.5.1 Concepts and terminology of hypothesis testing

5.5.2 Tests for binomial, Poisson, and normal means

5.5.3 Type I and Type II errors and their probabilities

6. Pure Mathematics 3

6.1 Algebra

6.1.1 Modulus equations and inequalities

6.1.2 Polynomial division and factor theorem

6.1.3 Partial fractions and decomposition

6.1.4 Binomial expansion for rational indices and validity of expansion

6.2 Logarithmic and exponential functions

6.2.1 Properties of logarithms and exponents

6.2.2 Solving equations and transforming to linear form

6.3 Trigonometry

6.3.1 Graphs and properties of all six trigonometric functions

6.3.2 Advanced identities and trigonometric expansions

6.4 Differentiation

6.4.1 Derivatives including tan–1 x and composite functions

6.4.2 Product, quotient, parametric and implicit differentiation

6.5 Integration

6.5.1 Standard and advanced integrals including sec², partial fractions and rational functions

6.5.2 Integration by parts and substitution

6.6 Numerical solution of equations

6.6.1 Root approximation and iteration methods

6.7 Vectors

6.7.1 Vector operations, equations of lines, and intersection

6.7.2 Scalar product, angles and perpendicular distances

6.8 Differential equations

6.8.1 Formulating and solving first-order separable equations

6.8.2 Using initial conditions and interpreting solutions

6.9 Complex numbers

6.9.1 Cartesian form, operations, and Argand diagram

6.9.2 Polar form, roots, multiplication and division

6.9.3 Loci and geometric interpretation of complex numbers

Basic rules and chain rule for differentiation

Topic 2/3

Revision Notes
Flashcards
Past Paper Analysis
Questions
Videos

Your Flashcards are Ready!

15 Flashcards in this deck.

Basic Rules and Chain Rule for Differentiation

Introduction

Differentiation is a cornerstone of calculus, enabling the analysis of how functions change with respect to their variables. In the context of the AS & A Level Mathematics course (9709), understanding the basic rules and the chain rule for differentiation is essential for solving a wide range of mathematical problems. This article explores these fundamental concepts in depth, providing clear explanations, practical examples, and insights to enhance your mastery of differentiation.

Key Concepts

Understanding Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function's output changes as its input changes. Formally, the derivative of a function $ f(x) $ at a point $ x $ is defined as:

$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$

This limit, if it exists, represents the slope of the tangent line to the function at the point $ x $, indicating the instantaneous rate of change.

Basic Differentiation Rules

Mastering the basic rules of differentiation is crucial for efficiently computing derivatives of various functions. The primary rules include:

Power Rule: For any real number $ n $, the derivative of $ f(x) = x^n $ is $ f'(x) = nx^{n-1} $.
Constant Multiple Rule: If $ f(x) = c \cdot g(x) $, where $ c $ is a constant, then $ f'(x) = c \cdot g'(x) $.
Sum and Difference Rules: For functions $ f(x) $ and $ g(x) $, the derivative of $ f(x) \pm g(x) $ is $ f'(x) \pm g'(x) $.
Product Rule: For the product of two functions $ f(x) $ and $ g(x) $, the derivative is $ f'(x)g(x) + f(x)g'(x) $.
Quotient Rule: For the quotient of two functions $ f(x) $ and $ g(x) $, the derivative is $ \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} $.

Power Rule

The Power Rule is one of the most frequently used rules in differentiation. It simplifies the process of finding derivatives of polynomial functions.

Formula:

$$ \frac{d}{dx}x^n = nx^{n-1} $$

Example: Differentiate $ f(x) = x^5 $.

Solution:

$$ f'(x) = 5x^{5-1} = 5x^4 $$>

Constant Multiple Rule

The Constant Multiple Rule allows you to factor out constants when differentiating functions.

Formula:

$$ \frac{d}{dx}[c \cdot g(x)] = c \cdot g'(x) $$>

Example: Differentiate $ f(x) = 7x^3 $.

Solution:

$$ f'(x) = 7 \cdot 3x^{3-1} = 21x^2 $$>

Sum and Difference Rules

The Sum and Difference Rules enable the differentiation of functions that are added or subtracted.

Formulas:

$$ \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) $$> $$ \frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x) $$>

Example: Differentiate $ f(x) = x^2 + 3x $.

Solution:

$$ f'(x) = 2x + 3 $$>

Product Rule

The Product Rule is essential when differentiating the product of two functions.

Formula:

$$ \frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) $$>

Example: Differentiate $ f(x) = x^2 \cdot \sin(x) $.

Solution:

$$ f'(x) = 2x \cdot \sin(x) + x^2 \cdot \cos(x) $$>

Quotient Rule

The Quotient Rule is used when differentiating the ratio of two functions.

Formula:

$$ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} $$>

Example: Differentiate $ f(x) = \frac{x^3}{\cos(x)} $.

Solution:

$$ f'(x) = \frac{3x^2 \cdot \cos(x) - x^3 \cdot (-\sin(x))}{\cos^2(x)} = \frac{3x^2 \cos(x) + x^3 \sin(x)}{\cos^2(x)} $$>

Chain Rule

The Chain Rule is a powerful tool for differentiating composite functions, where one function is nested inside another.

Formula:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$>

In the context of functions, if $ y = f(u) $ and $ u = g(x) $, then:

$$ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) $$>

Example: Differentiate $ f(x) = (3x + 2)^4 $.

Solution:

Let $ u = 3x + 2 $, then $ f(x) = u^4 $.

Differentiate $ f $ with respect to $ u $:

$$ \frac{df}{du} = 4u^3 $$>

Differentiate $ u $ with respect to $ x $:

$$ \frac{du}{dx} = 3 $$>

Apply the Chain Rule:

$$ \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} = 4u^3 \cdot 3 = 12(3x + 2)^3 $$>

Composite Example Utilizing Multiple Rules

Example: Differentiate $ f(x) = (2x^3 + x)^5 \cdot e^{x^2} $.

Solution:

This requires applying both the Product Rule and the Chain Rule.

Let $ u = (2x^3 + x)^5 $ and $ v = e^{x^2} $.

Differentiate $ u $ using the Chain Rule:

$$ \frac{du}{dx} = 5(2x^3 + x)^4 \cdot (6x^2 + 1) $$>

Differentiate $ v $:

$$ \frac{dv}{dx} = e^{x^2} \cdot 2x $$>

Apply the Product Rule:

$$ f'(x) = \frac{du}{dx} \cdot v + u \cdot \frac{dv}{dx} = 5(2x^3 + x)^4 (6x^2 + 1)e^{x^2} + (2x^3 + x)^5 \cdot 2x e^{x^2} $$>

Advanced Concepts

Theoretical Explanations and Mathematical Derivations

The Chain Rule is not just a procedural tool but is deeply rooted in the foundational principles of calculus. To understand its theoretical underpinnings, consider the composition of two differentiable functions $ f $ and $ g $, where $ y = f(u) $ and $ u = g(x) $. The derivative of $ y $ with respect to $ x $ can be derived as follows:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$>

This derivation leverages the limit definition of the derivative and the concept of function composition, ensuring the rule's validity across various differentiable functions.

Higher-Order Derivatives and the Chain Rule

Applying the Chain Rule multiple times facilitates the computation of higher-order derivatives for composite functions. For instance, to find the second derivative of $ f(x) = (2x^2 + 3x + 1)^4 $, follow these steps:

First Derivative:

$$ f'(x) = 4(2x^2 + 3x + 1)^3 \cdot (4x + 3) $$>

Second Derivative:

$$ f''(x) = 4 \cdot 3(2x^2 + 3x + 1)^2 \cdot (4x + 3)^2 + 4(2x^2 + 3x + 1)^3 \cdot 4 $$>

This example illustrates how the Chain Rule integrates with other differentiation techniques to handle more complex functions.

Implicit Differentiation

Implicit differentiation deals with equations where the dependent and independent variables are not explicitly solved for one another. The Chain Rule plays a crucial role in this process.

Example: Differentiate $ x^2 + y^2 = 25 $ with respect to $ x $.

Solution:

Differentiating both sides:

$$ 2x + 2y \cdot \frac{dy}{dx} = 0 $$>

Solving for $ \frac{dy}{dx} $:

$$ \frac{dy}{dx} = -\frac{x}{y} $$>

This technique is invaluable when dealing with circles, ellipses, and other implicit curves.

Logarithmic Differentiation

Logarithmic differentiation simplifies the differentiation of functions that are products or quotients of multiple terms, especially when raised to variable exponents.

Example: Differentiate $ f(x) = \frac{(x^2 + 1)^5}{(x^3 - x + 2)^4} $.

Solution:

Take the natural logarithm of both sides:

$$ \ln f(x) = 5 \ln(x^2 + 1) - 4 \ln(x^3 - x + 2) $$>

Differentiate implicitly:

$$ \frac{f'(x)}{f(x)} = \frac{10x}{x^2 + 1} - \frac{12x^2 - 4}{x^3 - x + 2} $$>

Solve for $ f'(x) $:

$$ f'(x) = f(x) \left( \frac{10x}{x^2 + 1} - \frac{12x^2 - 4}{x^3 - x + 2} \right ) $$>

Substitute $ f(x) $ back:

$$ f'(x) = \frac{(x^2 + 1)^5}{(x^3 - x + 2)^4} \left( \frac{10x}{x^2 + 1} - \frac{12x^2 - 4}{x^3 - x + 2} \right ) $$>

Applications in Physics and Engineering

The Chain Rule is extensively used in physics and engineering to model and analyze systems where variables are interdependent.

Example: Determine the acceleration of an object whose position is given by $ s(t) = (3t^2 + 2t + 1)^3 $.

Solution:

First, find the velocity $ v(t) = \frac{ds}{dt} $ using the Chain Rule:

$$ v(t) = 3(3t^2 + 2t + 1)^2 \cdot (6t + 2) = 3(3t^2 + 2t + 1)^2 (6t + 2) $$>

Next, find the acceleration $ a(t) = \frac{dv}{dt} $ by applying the Chain Rule again:

$$ a(t) = 3 \left[ 2(3t^2 + 2t + 1)(6t + 2)^2 + (3t^2 + 2t + 1)^2 \cdot 6 \right ] $$>

This demonstrates how the Chain Rule facilitates the computation of higher-order derivatives in dynamic systems.

Interdisciplinary Connections

Differentiation, and particularly the Chain Rule, connects to various fields beyond pure mathematics.

Economics: Analyzing marginal costs and revenues involves differentiation of profit functions.
Biology: Modeling population growth rates requires understanding how different factors change over time.
Computer Science: Optimization algorithms in machine learning utilize derivatives to minimize error functions.

By mastering the Chain Rule, students can apply mathematical principles to solve real-world problems across diverse disciplines.

Complex Problem-Solving

The integration of multiple differentiation rules, including the Chain Rule, empowers students to tackle complex, multi-step problems.

Example: Differentiate $ f(x) = \ln((5x^2 - 3x + 2)^4) $.

Solution:

First, simplify using logarithmic properties:

$$ \ln((5x^2 - 3x + 2)^4) = 4 \ln(5x^2 - 3x + 2) $$>

Differentiate using the Chain Rule:

$$ f'(x) = 4 \cdot \frac{1}{5x^2 - 3x + 2} \cdot (10x - 3) = \frac{4(10x - 3)}{5x^2 - 3x + 2} $$>

This example illustrates the application of multiple differentiation techniques in solving a single problem.

Comparison Table

Rule	Definition	Formula
Power Rule	Differentiates functions of the form $ x^n $	$ \frac{d}{dx}x^n = nx^{n-1} $
Product Rule	Differentiates the product of two functions	$ (fg)' = f'g + fg' $
Quotient Rule	Differentiates the quotient of two functions	$ \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} $
Chain Rule	Differentiates composite functions	$ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) $

Summary and Key Takeaways

Differentiation is essential for analyzing the rate of change in mathematical functions.
The Power, Constant Multiple, Sum/Difference, Product, and Quotient Rules form the foundation of basic differentiation techniques.
The Chain Rule is crucial for differentiating composite functions, enabling the solution of more complex problems.
Advanced applications of differentiation extend to various disciplines, including physics, engineering, economics, and computer science.
Mastery of these rules enhances problem-solving skills and prepares students for higher-level mathematical challenges.

Examiner Tip

Tips

To master the Chain Rule, practice identifying composite functions by spotting nested operations. A useful mnemonic is "Inside-Out," reminding you to differentiate the outer function first, then multiply by the derivative of the inner function. Additionally, always simplify your expressions before differentiating to make the application of the Chain Rule more straightforward and reduce the likelihood of errors during calculations.

Did You Know

The Chain Rule isn't just a mathematical concept; it's fundamental in various scientific breakthroughs. For instance, in neuroscience, it helps model how signals propagate through complex neural networks. Additionally, the Chain Rule plays a crucial role in the development of animation software, allowing for the smooth transformation of objects by calculating the necessary motion paths. Understanding these applications highlights the versatile nature of differentiation in real-world scenarios.

Common Mistakes

Students often confuse the Chain Rule with other differentiation rules, leading to incorrect results. For example, mistakenly applying the Product Rule instead of the Chain Rule when differentiating $ f(x) = \sin(3x) $ results in errors. Another common mistake is forgetting to multiply by the derivative of the inner function. To avoid these pitfalls, always identify the outer and inner functions clearly before applying the Chain Rule.

FAQ

What is the Chain Rule in differentiation?

The Chain Rule is a fundamental formula in calculus used to find the derivative of composite functions. It states that the derivative of a composite function $ f(g(x)) $ is $ f'(g(x)) \cdot g'(x) $.

When should I use the Chain Rule?

Use the Chain Rule when differentiating functions composed of other functions, such as $ \sin(3x) $ or $ (2x + 1)^5 $. It's essential for handling nested operations in complex expressions.

Can the Chain Rule be applied to higher-order derivatives?

Yes, the Chain Rule can be applied multiple times to find higher-order derivatives of composite functions. Each application requires differentiating the outer function and multiplying by the derivative of the inner function successively.

What is the difference between the Chain Rule and the Product Rule?

The Chain Rule is used for differentiating composite functions, whereas the Product Rule is used for differentiating the product of two separate functions. Confusing the two can lead to incorrect derivatives.

Are there functions where the Chain Rule does not apply?

The Chain Rule applies to any pair of functions that are differentiable and composed together. However, if a function is not differentiable at a certain point, the Chain Rule cannot be applied there.

1. Mechanics

1.1 Forces and equilibrium

1.1.1 Identifying and resolving forces

1.1.2 Equilibrium of particles and friction

1.1.3 Normal and frictional components of contact forces

1.1.4 Coefficient of friction and limiting equilibrium

1.1.5 Application of Newton’s third law

1.2 Kinematics of motion in a straight line

1.2.1 Scalar and vector quantities in motion

1.2.2 Displacement-time and velocity-time graphs

1.2.3 Calculus in kinematics

1.2.4 Constant acceleration equations