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Differentiation I

Modelling with differentiation

Modelling with differentiation

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Further kinematics

Vector methods with projectiles

Variable acceleration in one dimension

Differentiating vectors

Integrating vectors

Application of forces

Static particles

Modelling with statics

Friction with static particles

Statics of rigid bodies

Dynamics and inclined planes

Connected particles

Projectiles

Horizontal projection

Projectiles: Horizontal and vertical components

Projection at any angle

Equation of the trajectory of a projectile

Forces and friction

Resolving forces

Inclined planes

The coefficient of friction

Moments

Moments

Resultant moments

Equilibrium

Centre of mass

Tilting

Laminas

Forces and motion

Force diagrams

Forces as vectors

Forces and acceleration

Motion in two dimensions

Connected particles

Pulleys

Variable acceleration

Functions of time: Displacement, velocity, acceleration

Using differentiation to find velocity and acceleration

Maxima and minima problems

Using integration to find velocity and displacement

Constant acceleration formulae

Constant acceleration

Displacement-time graphs

Velocity-time graphs

Constant acceleration: Deriving formulae

Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

Modelling assumptions

Quantities and units in mechanics

Working with vectors

Normal distribution

The normal distribution

The normal distribution: Finding probabilities

The inverse normal distribution function

The standard normal distribution

Finding the mean and standard deviation

Normal approximation to the binomial distribution

Hypothesis testing with the normal distribution

Conditional probability

Set notation

Conditional probability

Conditional probability in Venn diagrams

Probability formulae

Conditional probability: Tree diagrams

Correlation and hypothesis testing

Exponential models

Measuring correlation

Hypothesis testing for zero correlation

Hypothesis testing I

Hypothesis testing

Hypothesis testing: Finding critical values

One-tailed tests

Two-tailed tests

Binomial distribution

Probability distributions

The binomial distribution

Cumulative probabilities

Probability

Calculating probabilities

Venn diagrams

Probability: Mutually exclusive and independent events

Probability: Tree diagrams

Representation of data

Outliers

Box plots

Cumulative frequency

Histograms

Comparing data sets

Correlation

Linear regression

Measures of location and spread

Measures of central tendency: Mean, median, mode

Measures of spread: Range

Variance and standard deviation

Data collection

Population and samples

Sampling techniques

Types of data

Vectors II

3D coordinates

Vectors in 3D

Solving geometric problems

Applications to mechanics: 3D vectors

Numerical methods

Locating roots

Iteration

Staircase and cobweb diagrams

Newton-Raphson method

Numerical methods: Applications to modelling

Integration II

Integrating standard functions

Integrating f(ax + b)

Integration with trigonometric identities

Reverse chain rule

Integration by substitution

Integration by parts

Integration using partial fractions

Finding areas using integration

The trapezium rule

Solving differential equations

Modelling with differential equations

Differentiation II

Differentiating sin x and cos x

Differentiating exponentials and logarithmic functions

The chain rule

The product rule

The quotient rule

Differentiating inverse functions

Differentiating trigonometric functions

Parametric differentiation

Implicit differentiation

Second derivatives: Concave and convex functions

Connected rates of change

Trigonometry and modelling

Addition formulae

Using the angle addition formulae

Double angle formulae

Solving trigonometric equations: Double angle formula

Simplifying a cos x + b sin x

Proving trigonometric identities

Modelling with trigonometric functions

Trigonometric functions

Secant, cosecant and cotangent

Graphs of sec x, cosec x and cot x

Solving equations with sec x, cosec x and cot x

Further trigonometric identities

Inverse trigonometric functions

Radians

Radian measure

Arc length

Areas of sectors and segments

Solving trigonometric equations

Small angle approximations

Sequences and series

Sequences revision

Arithmetic sequences

Arithmetic series

Geometric sequences

Geometric series

Sum to infinity

Sigma notation

Modelling with sequences and series

Binomial expansion II

Binomial expansion: (1+x)^n

Binomial expansion: (a+bx)^n

Binomial expansion with partial fractions

Binomial expansion of compound expressions

Parametric equations

Parametric equations

Using trigonometric identities

Sketching parametric curves

Points of intersection of parametric curves

Modelling with parametric equations

Functions

Functions and mappings

Composite functions

Inverse functions

The modulus function

y = |f(x)| and y = f(|x|)

Combining transformations

Solving modulus equations

Modulus inequalities

Algebraic fractions

Operations with algebraic fractions

Partial fractions

Repeated factors

Algebraic division

Proof II

Proof by contradiction

Criticising proofs

Vectors I

Vectors

Representing vectors

Magnitude and direction of a vector

Position vectors and displacement vectors

Solving geometric problems

Modelling with vectors

Integration I

Integrating x^n

Indefinite integrals

Finding functions by integrating

Definite integrals

Area under a curve

Area under the x-axis

Area between a curve and a line

Differentiation I

Finding the gradient of a curve

Differentiation from first principles

Differentiating x^n

Differentiating functions with multiple terms

Gradients, tangents and normals

Increasing and decreasing functions

Second order derivatives

Stationary points

Sketching gradient functions

Modelling with differentiation

Exponentials and logarithms

Exponential functions

y = e^x

Exponential modelling

Logarithms

Laws of logarithms

Solving equations using logarithms

Natural logarithms

Logarithms and non-linear data

Harder exponential modelling

Trigonometric equations

Angles in all four quadrants

Exact trigonometric values

Trigonometric identities

Simple trigonometric equations

Harder trigonometric equations

Equations and identities

Trigonometry

The cosine rule

The sine rule

Area of a triangle

Solving triangle problems

Trig graphs: sin x, cos x, tan x

Transforming trigonometric graphs

Binomial expansion I

Pascal's triangle

Factorial notation

Binomial expansion

Solving binomial problems

Binomial estimation

Circles

Midpoints and perpendicular bisectors

Equation of a circle

Intersection of straight lines and circles

Using tangent and chord properties

Circles and triangles

Intersection of two circles

Straight line graphs

y = mx + c

Equations of straight lines

Parallel and perpendicular lines

Length and area

Modelling with straight lines

Transformations

Translating graphs

Stretching and reflecting graphs

Transforming graphs

Sketching graphs

Cubic graphs

Quartic graphs

Reciprocal graphs

Points of intersection

Using the discriminant

Direct and inverse proportion

Polynomials

Functions

Expanding brackets

Factorising

Simplifying algebraic fractions

Polynomial division

The factor theorem

Equations and inequalities

Linear simultaneous equations

Quadratic simultaneous equations

Set notation and interval notation

Linear inequalities

Quadratic inequalities

Inequalities on graphs

Inequality regions

Quadratics

Solving quadratic equations

Completing the square

Quadratic graphs

Identifying the discriminant

Solving disguised quadratics

Indices and surds

Laws of indices

Negative and fractional indices

Surds

Rationalising the denominator

Proof I

Mathematical structure and arguments

Disproof by counterexample

Proof by deduction

Proof by exhaustion

Explainer Video

Tutor: Toby

Summary

Modelling with differentiation

​​In a nutshell

Differentiating an equation finds the gradient of the equation's curve. This represents the rate of change of one variable with respect to the other. Using differentiation in real-life scenarios can show how variables change, for example, over time.



A look at notation

The "d\text dd" in dydx\frac{\text dy}{\text dx}dxdy​ can be thought of as meaning "a small change in". This means that, dydx=a small change in y÷a small change in x\dfrac{\text dy}{\text dx}=\text{a small change in }y \div\text{a small change in }xdxdy​=a small change in y÷a small change in x ​which is the rate of change of yyy with respect to xxx. 


Now suppose yyy and xxx were replaced with variables that represented some real-life quantities. The derivative would now represent the rate of change of the yyy-replacement with respect to the xxx-replacement.


Example 1

The equation for the area of a circle, AAA, is given in terms of its radius, rrr:

A=πr2A=\pi r^2A=πr2​​


Find an equation for the rate of change of the area with respect to the radius.


This is given by the first derivative of AAA with respect to rrr:

dAdr=2πr‾\underline{\dfrac{\text dA}{\text dr}=2\pi r}drdA​=2πr​​​


​​Example 2

A cuboid is being filled with water from a tap. The volume in cm3\text{cm}^3cm3​ of water in the cuboid is given by V=5t2V=5t^2V=5t2 where ttt is the time in seconds since the tap was turned on. It takes ten seconds to fill the cuboid. Find the rate of change of the volume with respect to time after three seconds.


The rate of change is found by differentiating VVV with respect to ttt:

dVdt=10t\dfrac{\text dV}{\text dt}=10tdtdV​=10t​​


Find the rate of change when t=3t=3t=3:

  dVdt=10(3)=30 cm3s−1‾\dfrac{\text dV}{\text dt}=10(3)=\underline{30 \text{ cm}^3\text s^{-1}}dtdV​=10(3)=30 cm3s−1​. 


Note: The unit has come from the fact that you have a volume unit (cm3\text{cm}^3cm3) per time unit (s\text ss).



Optimisation

Models of real-life scenarios can help you maximise or minimise different quantities. Recall that maxima and minima have zero gradients. They are classified using the second derivative. 


Example 3

A cuboid has two perpendicular sides of length xxx metres and the third perpendicular side of length yyy metres. You have that the total surface area of the cuboid is 54 m254\text{ m}^254 m2. Given that the value xxx can vary, find the maximum volume of the cuboid. Justify why it is the maximum. 


Start with the equation for the volume of a cuboid: 

​Vcuboid=height×base×lengthV_{\text{cuboid}}=\text{height}\times\text{base}\times\text{length}Vcuboid​=height×base×length​​


Hence for this example you have:

V=x2yV=x^2yV=x2y​​


Now the surface area:

Acuboid=2(base×height)+2(length×height)+2(base×length)A_{\text{cuboid}}=2(\text{base}\times \text{height})+2(\text{length}\times \text{height})+2(\text{base}\times \text{length})Acuboid​=2(base×height)+2(length×height)+2(base×length)​​


Hence for this example you have:

54=2x2+4xy 27=x2+2xy54=2x^2+4xy\\\space\\27=x^2+2xy54=2x2+4xy 27=x2+2xy​​


Given that xxx can vary, and you are looking for how volume changes, use these equations to find an equation for VVV​ in terms of xxx:

27=x2+2xy y=27−x2x=27x−x27=x^2+2xy\\\space\\y=\dfrac{27-x^2}{x}=\frac{27}x-x27=x2+2xy y=x27−x2​=x27​−x​​


Insert this into the equation for volume:

V=x2y V=x2(27x−x)=27x−x3V=x^2y\\\space\\V=x^2\bigg(\dfrac{27}x-x\bigg)=27x-x^3V=x2y V=x2(x27​−x)=27x−x3​​


Differentiate VVV with respect to xxx:

dVdx=27−3x2\dfrac{\text dV}{\text dx}=27-3x^2dxdV​=27−3x2​​


Set this equal to zero and solve for xxx:

0=27−3x2 3x2=27 x2=9 x=±30=27-3x^2\\\space\\3x^2=27\\\space\\x^2=9\\\space\\x=\pm30=27−3x2 3x2=27 x2=9 x=±3​​


Since xxx is a measurement of distance, it follows that it must be positive, so x=3x=3x=3.

To confirm that this represents a maximum VVV, find the second derivative and insert x=3x=3x=3 to get a maximum:

d2Vdx2=−6x d2Vdx2=−6(3)=−18<0\dfrac{\text d^2V}{\text dx^2}=-6x\\\space\\\dfrac{\text d^2V}{\text dx^2}=-6(3)=-18<0dx2d2V​=−6x dx2d2V​=−6(3)=−18<0​​


Thus when x=3x=3x=3, VVV is at a maximum. Insert this xxx into the equation for VVV:

V=30x−x3=30(3)−(3)3=63V=30x-x^3=30(3)-(3)^3=63V=30x−x3=30(3)−(3)3=63​​


The maximum volume is 63 m3‾\underline{63 \text{ m}^3}63 m3​.



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FAQs - Frequently Asked Questions

What does the first derivative represent?

This represents the rate of change of one variable with respect to the other.

How do you interpret a model using differentiation?

Consider dy/dy. Now suppose y and x were replaced with variables that represented some real-life quantities. The derivative would now represent the rate of change of the y-replacement with respect to the x-replacement.

What is optimisation?

Optimisation is about finding the maximum or minimum of some variable. Models of real-life scenarios can help you maximise or minimise different quantities. Recall that maxima and minima have zero gradients. They are classified using the second derivative.

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