My Math Blog

 The first rule, the Power Rule, is the rule used upon functions of x that are bases for non-variable values. The second rule, the Reciprocal Rule, is used on the reciprocal of certain functions or identities, however this rule can be derived (aka found from) the Power Rule. A reciprocal is merely 1 divided by a function, this can be rewritten as f(x)^(-1), thus the function becomes the base of the power of "-1". The third rule, the product rule, is the rule used on the products of multiple functions and identities. This rule is based off the initial theory used to formulate differentiation, the use of the difference of hypothetical infinitesimally small differences. As the derivative is the change, we only consider the two blue boxes of uv' and u'v as a result of the change, we can ignore the u'v' since both values are small, we can consider it negligible. The fourth rule, the Quotient Rule, derivable from the Product Rule The fifth rule, the Chain rule, a si...

There are many "Rules" of differentiation, but in a sense these are shorthand which skip over the complicated proof that show why they work. Although, at the level of IGCSE these proofs are unnecessary to be placed in each question they are used, and can be assumed to be true when used, enough paper is used as it is. However, I do encourage to look and attempt to understand the proof, as it is a useful method for remembering the how and when to use them. The proof of each can be seen through the link to the eventual page for it:  Here The first rule, is known as the Power Rule, since it deals with functions of x by the power of a non-variable value. For example: f(x)^n. Can be differentiated by reducing the power by 1, and multiplying by the power and the derivative of f(x). The second rule, is known as the Reciprocal Rule, since the reciprocal of f(x) is 1/f(x) or (f(x))^(-1). In fact, this rule is formed by the Power Rule, however remembering this as it is, is fine, The thi...

Differentiation is a method to find the "Rate of Change" or "Change Per Unit Time" of an equation, this can mean the change in distance on a journey per hour, the change in volume of water in a leaking tub per minute, or even the change in distance of an accelerating race car per second. These results can be expanded further: finding the velocity of an accelerating ball at a certain distance, finding the current change in volume of a tub of water at a specific height, the finding the maximum area of an object with a constant perimeter, and more. As long as there is a change in a measurement that correlates to the change of another measurement, differentiation can be of use. First and foremost, the differentiated result of a function is known as a derivative. For example, speed can be considered the "Change of Distance Per Unit Time", so speed is the 'Derivative of Distance". Second of all, the amount of times something can be differentiated is inf...

In this scenario, Point P lies on line AB, and ratio AP : PB is 1 : 3. However what does this mean? First, the ratio implies that PB is 3 times the length of AP,. Second, As Points A, P, and B, are all in a line, the direction of AB, AP and PB are the same. Third, seen in the diagram, Vectors OA and OB are known as Vectors "a" and "b" respectively. Therefore, by knowing the statements above, one can find OP. The first step of finding OP is what are it's closest or most relevant connections. Point P lies on line AB, and both OA and OB are known, thus OP will likely be a combination of OA + AP or OB + BP. Second, after finding a path towards OP, next is to find the Vector notation for AB, AP and/or PB. AB can be found by AO + OB = -a + b = b - a. Therefore AP  =  (b - a)/4, and PB = (3/4)(b - a). Thus OP can be found by OP = OA + AP  or OP = OB + BP = OB - PB. The first method would be, OP = a + (b - a)/4 = b/4 + 3a/4. The second method being, OP = b - (3/4)(b - a...

Where Vector means " determining the position of one point in space relative to another". Things to note: 1) While not always the case, Point O is usually considered the origin point, meaning that it's position in 2 dimensional space is (0,0). 2) Due to the limitations of digital text, any Vector going by the naming convention of XY where X and Y are points, implies a Vector Line from Point X to Point Y. Where it would usually look, for OA, like: 3) Vector Lines can be added together, the rule being that the end point of one must be the beginning point of the next, for example: OA + AB = OB, since OA is the vector from Point O to Point A, then AB is the vector from Point A to Point B, adding them together creates a vector that goes from Point O to Point A to Point B. 4) Vectors can also be denoted in a Variable-like fashion, for example OA can be denoted as "a". Thus these Vectors can be arranged in an equation to form the Vector of one point to another. For exa...

 For |ax+b| + c = d, one can square both sides to use the quadratic equation to get both answers of x. First, separate the absolute value from those that are not. Second, square both sides, since squaring anything ensures that the results are always positive when the values are real. This can be shown if |x| can be considered as √[(x)^2]. Thus we can extrapolate that, Therefore, So, We can simplify the Right Hand Side, From this point forward, the Quadratic Equation can be used to find the values of x.