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Condition If the coefficient of $x^2$ is 1, think of two numbers which: 1. Add to make the $b$ 2. Multiply to make $c$ Otherwise, utilise the cross method. Cross Method 1. Left sid...
Condition If the coefficient of $x^2$ is 1, think of two numbers which: 1. Add to make the $b$ 2. Multiply to make $c$ Otherwise, utilise the cross method. Cross Method 1. Left sid...
$y = 2$ is a constant, it is the same as $y = 2x^0$. $y = x + 2$ is linear. $y = x^2 + 2x 4$ is quadratic. The gradient of a linear line can be calculated by doing $\frac{\Delta y}...
$y = 2$ is a constant, it is the same as $y = 2x^0$. $y = x + 2$ is linear. $y = x^2 + 2x 4$ is quadratic. The gradient of a linear line can be calculated by doing $\frac{\Delta y}...
WARNING This document was AI generated. Surds are numbers left in square root or cube root form because their exact value cannot be written as a whole number or a fraction. They ar...
WARNING This document was AI generated. Surds are numbers left in square root or cube root form because their exact value cannot be written as a whole number or a fraction. They ar...
Implicit differentiation is a different way to find the derivative when $y$ is "stuck" inside an equation and you can't or don't want to isolate it as $y = \dots$. Always treat $y$...
$h x = \frac{1}{f x }$ $$ \begin{array}{0} \text{cosec} \space x = \frac{1}{\sin x} \\ \text{cosec}^2 \space x = \frac{1}{\sin^2 x} \\ \sec x = \frac{1}{\cos x} \\ \cot x = \frac{1...
$$ \begin{array}{0} \frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt} \\ \frac{dy}{dx} = \frac{dy}{dt} \times \frac{dt}{dx} \\ \frac{dy}{dx} = \frac{dy}{dx} \end{array} $$
$$ \begin{array}{0} y = \frac{u x }{v x } \\ \frac{dy}{dx} = \frac{v x u' x u x v' x }{v x ^2} \end{array} $$
$$ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} $$ Chain rule is used to decompose working with complex equations within differentiation much easier. It allows you to substit...
Exponential Rules $$ \begin{array}{0} \frac{d}{dx} e^x = e^x \\ \frac{d}{dx} e^{kx} = ke^{kx} \\ \frac{d}{dx} a^x = a^x \ln a \\ \frac{d}{dx} a^{kx} = a^{kx} k \ln a \end{array} $$...
$$ \begin{array}{0} y = uv \\ \frac{dy}{dx} = v \frac{du}{dx} + u \frac{dv}{dx} \end{array} $$ Examples $y = x \sin x$ $$ \begin{array}{0} y = x \sin x \\ u = x \\ v = \sin x \\ \f...
$$ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} $$ Just like you can get $x$ from $y$ by getting the reciprocal of $y$, you can do the same for $\frac{dx}{dy}$ , allowing you to easily...
Using Pascal's triangle or the binomial equation , you can quickly expand an equation such as $ x+h ^x$. To do this you: 1. Look at the $n$ th row of the triangle counting the very...
Using Pascal's triangle or the binomial equation , you can quickly expand an equation such as $ x+h ^x$. To do this you: 1. Look at the $n$ th row of the triangle counting the very...
Factor theorem is the process of validating whether a factor is a factor of an equation. A factor is a number where the a % b = 0 . For example, $ x + 3 $ is a factor of $f x = x^2...
Factor theorem is the process of validating whether a factor is a factor of an equation. A factor is a number where the a % b = 0 . For example, $ x + 3 $ is a factor of $f x = x^2...
To get the local turning point of a quadratic, you can either: 1. Complete the Square Completing The Square 2. Use differentiation anx Equation Remember that the $gradient = 0$ and...
To get the local turning point of a quadratic, you can either: 1. Complete the Square Completing The Square 2. Use differentiation anx Equation Remember that the $gradient = 0$ and...
NOTE These were also used for Grade 9 questions in GCSE. The Sine Rule and Cosine Rule are used to find unknown sides or angles in non right angled triangles. Sine Rule The Sine Ru...
NOTE These were also used for Grade 9 questions in GCSE. The Sine Rule and Cosine Rule are used to find unknown sides or angles in non right angled triangles. Sine Rule The Sine Ru...
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Vector vs. Scaler A vector has both a magnitude and direction , whereas a scaler has only a magnitude .
A unit vector is any vector with a magnitude of one. Conversion To convert a vector to a unit vector, you need to: 1. Calculate the magnitude using the Pythagoras theorem 2. Divide...
A unit vector is any vector with a magnitude of one. Conversion To convert a vector to a unit vector, you need to: 1. Calculate the magnitude using the Pythagoras theorem 2. Divide...
TODO Write and explain techniques. Choose Example $$ \begin{array}{0} 1+ax ^10 = 1^10 + 1^9 {10\choose {1}} ax ^1 + 1^8 + {10\choose {2}} ax ^2 \dots \end{array} $$ Year 13 Techniq...
TODO Write and explain techniques. Choose Example $$ \begin{array}{0} 1+ax ^10 = 1^10 + 1^9 {10\choose {1}} ax ^1 + 1^8 + {10\choose {2}} ax ^2 \dots \end{array} $$ Year 13 Techniq...
Limits can be represented as either $ x ^a {b}$ or $\int^a {b}x$. The limits represent both the upper and lower bounds of a definite integral the interval over which to calculate t...
Limits can be represented as either $ x ^a {b}$ or $\int^a {b}x$. The limits represent both the upper and lower bounds of a definite integral the interval over which to calculate t...
WARNING This document was partially AI generated, however adapted. $e$ is used as a base because it has the unique property where the gradient of $y = e^x$ is equal to $e^x$ itself...
WARNING This document was partially AI generated, however adapted. $e$ is used as a base because it has the unique property where the gradient of $y = e^x$ is equal to $e^x$ itself...
Integration is the reverse of Differentiation. It uses the formula $y = \frac{k}{n+1}x^{n+1} + c$ as long as $n \not= 1$. It can be summarised in three steps: 1. Add one to the pow...
Integration is the reverse of Differentiation. It uses the formula $y = \frac{k}{n+1}x^{n+1} + c$ as long as $n \not= 1$. It can be summarised in three steps: 1. Add one to the pow...
A logarithm is the exponent to which a base must be raised to produce a given number. Equation $$ \begin{array}{0} a^x = b \\ x = \log {a}b \\ \end{array} $$ Usage Rules You cannot...
A logarithm is the exponent to which a base must be raised to produce a given number. Equation $$ \begin{array}{0} a^x = b \\ x = \log {a}b \\ \end{array} $$ Usage Rules You cannot...
A stationary point on a curve is where the gradient is $0$. This occurs when $f' x = 0$. For example, the curve $y = x^2$ has a stationary point at $ 0, 0 $. Local Maximum and Mini...
A stationary point on a curve is where the gradient is $0$. This occurs when $f' x = 0$. For example, the curve $y = x^2$ has a stationary point at $ 0, 0 $. Local Maximum and Mini...
WARNING This document was AI generated. The equation $ax^n$ is the standard form used for differentiating terms in a polynomial. The Power Rule To differentiate $ax^n$, you multipl...
WARNING This document was AI generated. The equation $ax^n$ is the standard form used for differentiating terms in a polynomial. The Power Rule To differentiate $ax^n$, you multipl...
WARNING This does not have any practical example. It is just a demonstration. See Binomial Expansion for technique. $$ \begin{array}{0} x + 2y ^2 = 1x^3 2y ^0 + 3x^2 2y + 3x 2y ^2...
WARNING This does not have any practical example. It is just a demonstration. See Binomial Expansion for technique. $$ \begin{array}{0} x + 2y ^2 = 1x^3 2y ^0 + 3x^2 2y + 3x 2y ^2...
$ x a ^2 + y b ^2 = r^2$ $ a, b $ is the centre of the circle. $ x, y $ is the current position. $r$ is the radius of the circle. Example: Finding the equation of a circle Given: c...
$ x a ^2 + y b ^2 = r^2$ $ a, b $ is the centre of the circle. $ x, y $ is the current position. $r$ is the radius of the circle. Example: Finding the equation of a circle Given: c...
$$ \begin{array}{1} a^\frac{1}{m} = \sqrt m { a } \\ a^\frac{n}{m} = \sqrt m { a^n } \\ a^ m = \frac{1}{a^m} \\ a^0 = 1 \end{array} $$
$$ \begin{array}{1} a^\frac{1}{m} = \sqrt m { a } \\ a^\frac{n}{m} = \sqrt m { a^n } \\ a^ m = \frac{1}{a^m} \\ a^0 = 1 \end{array} $$
Formula This formula calculates the midpoint of both the $x$ and $y$ axes individually. $$ \begin{array}{1} \left \frac{x {1} + x {2}}{2}, \frac{y {1} + y {2}}{2} \right \end{array...
Formula This formula calculates the midpoint of both the $x$ and $y$ axes individually. $$ \begin{array}{1} \left \frac{x {1} + x {2}}{2}, \frac{y {1} + y {2}}{2} \right \end{array...
NOTE Revise. $$ a x + h ^2 + k $$ To get the co ordinates of the maximum/minimum turning point. $ x + h = 0$ or $ h$ provides the $x$ co ordinate. $k$ provides the $y$ co ordinate....
NOTE Revise. $$ a x + h ^2 + k $$ To get the co ordinates of the maximum/minimum turning point. $ x + h = 0$ or $ h$ provides the $x$ co ordinate. $k$ provides the $y$ co ordinate....
WARNING This document was AI generated. Equation $$ f' x = \lim {h \to 0} \frac{f x+h f x }{h} $$ Examples Function: $x^2$ Step 1: Find $f x+h $ $$ f x+h = x+h ^2 = x^2 + 2xh + h^2...
WARNING This document was AI generated. Equation $$ f' x = \lim {h \to 0} \frac{f x+h f x }{h} $$ Examples Function: $x^2$ Step 1: Find $f x+h $ $$ f x+h = x+h ^2 = x^2 + 2xh + h^2...
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$$ m = \frac{\Delta y}{\Delta x} $$
$$ m = \frac{\Delta y}{\Delta x} $$
To find the length of a line, you can use Pythagoras' Theorem. $$ A^2 = B^2 + C^2 $$ as: $$ d = \sqrt{ x^2 x^1 ^2 + y^2 y^2 ^2 } $$
To find the length of a line, you can use Pythagoras' Theorem. $$ A^2 = B^2 + C^2 $$ as: $$ d = \sqrt{ x^2 x^1 ^2 + y^2 y^2 ^2 } $$
Parallel lines are two lines which have the same gradient but never touch. Example $$ \begin{array}{1} y {1} = 5x + 2 \\ y {2} = 5x + 4 \end{array} $$ These linear lines are parall...
Parallel lines are two lines which have the same gradient but never touch. Example $$ \begin{array}{1} y {1} = 5x + 2 \\ y {2} = 5x + 4 \end{array} $$ These linear lines are parall...
Perpendicular lines are two lines which meet at 90°. If a line has the gradient of $m$, a line perpendicular will have the gradient of $ \frac{1}{m}$. If two lines are perpendicula...
Perpendicular lines are two lines which meet at 90°. If a line has the gradient of $m$, a line perpendicular will have the gradient of $ \frac{1}{m}$. If two lines are perpendicula...
Sine and Cosine $\sin x $ and $\cos x $ are periodic functions with a period of 360° or $2\pi$ radians . Both sine and cosine graphs oscillate between 1 and 1. The sine graph start...
Sine and Cosine $\sin x $ and $\cos x $ are periodic functions with a period of 360° or $2\pi$ radians . Both sine and cosine graphs oscillate between 1 and 1. The sine graph start...