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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...
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...
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Vector vs. Scaler A vector has both a magnitude and direction , whereas a scaler has only a magnitude .
To convert a vector to a unit vector, you need to: 1. Calculate the magnitude using the Pythagoras theorem 2. Divide both axes with the magnitude Demonstration TODO: Write iframe c...
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...
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...
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...
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. $$ \begin{array}{0} x + 2y ^2 = 1x^3 2y ^0 + 3x^2 2y + 3x 2y ^2...
$ x a ^2 + y b ^2 = r^2$ $ a, b $ is the centre of the circle. $ x, y $ is the current position. $r$ is the radius of the circle. Example: Finding the equation of a circle Given: c...
$$ \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} $$
Formula This formula calculates the midpoint of both the $x$ and $y$ axes individually. $$ \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....
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} $$
To find the length of a line, you can use Pythagoras' Theorem. $$ 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...
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...