§ 01The Core Belief
Mathematics is not a collection of formulas to memorise. It is a collection of ideas to understand. Formulas follow automatically once the idea is clear.
The difference between a student who remembers everything and one who forgets everything is not intelligence, it is understanding. When you understand why the derivative of tan x is sec²x, you will never confuse it with csc²x. When you understand why the inverse trig derivative uses a triangle, you will never need to memorise the six formulas by rote. When you understand why partial fractions works, you will know exactly what to write for each type of denominator without having to look it up.
This site was built on one conviction: if a student is struggling with a mathematical concept, the problem is almost always the explanation: not the student. Most textbooks are written by people who have understood the material for so long that they have forgotten what it felt like not to. They skip steps. They assume notation is obvious. They present results without motivation. This site tries to do the opposite.
§ 02The Four Specific Methods
These are the four techniques that make this site different from a standard textbook. Each one replaces memorisation with a logical procedure.
Before differentiating any trig function other than sin or cos, rewrite it entirely in terms of sin and cos. No exceptions.
Once rewritten, combine all terms over a single common denominator before applying the quotient rule. This prevents chaos in the algebra.
Any trig function starting with C (cos, cot, csc) produces a negative derivative. The C-Rule is the shortcut that describes the pattern in the results.
To differentiate arcsin, arccos or arctan, swap to the regular trig form, draw a right triangle, label it with Pythagoras, then differentiate implicitly.
Why Write Everything in sin and cos?
There are six trig functions (sin, cos, tan, cot, sec, csc) but only two of them are fundamental. The other four are all ratios built from sin and cos. By rewriting everything in terms of sin and cos before differentiating, you reduce six potentially confusing functions down to two that you already know how to differentiate. The quotient rule does the rest. This approach is consistent, reliable, and completely derivable from first principles, no formula sheet required.
Why the Triangle Method for Inverse Trig?
The standard textbook approach to finding d/dx[arcsin x] involves implicit differentiation followed by a step where you magically replace cos(arcsin x) with √(1−x²). Students are left wondering where that substitution came from. The triangle method makes it visible: draw the right triangle implied by the definition sin(y) = x, assign the sides using Pythagoras, and the third side √(1−x²) appears geometrically, no mystery, no magic, just a labelled triangle.
The Log e over Log e Trick
When differentiating log base a of x for any base a, the key insight is to convert to the natural logarithm using the change-of-base formula: log_a(x) = ln(x) ÷ ln(a). Since ln(a) is just a constant, it comes outside the derivative. The derivative of ln(x), which is 1÷x, does all the work. This technique handles every possible logarithm base without needing a separate formula for each one.
§ 03Who Made This
This site is written and checked by a small team, not generated at scale. Here's who's behind it.
A teacher at LM Mokoena High School, Mxolisi writes the maths explanations on this site and submits each one to Mr Mathebula, a maths tutor, for review before it's published.
A maths tutor who reviews every explanation Mxolisi writes before it goes live on the site, checking it for accuracy and clarity.
A Bachelor of Information Communication Technology graduate who has studied calculus, precalculus, probability and statistics. Iphesihle came up with the original idea for the site and wrote the basic maths problems it's built around.
A medical science student at the University of Cape Town, Lifa helped mainly with the algebra-based solutions used throughout the site.
A South African artist and rapper known as Flow T, recognised for his appearance on Isithembiso. Thando is the overall content checker for the site.
§ 04Who This Site Is For
This site is for any student who has ever stared at a maths page and thought: "I understand every word individually, but I have no idea what is going on."
Specifically, this site is built for high school students studying calculus for the first time, university students in first and second year mathematics, engineering, physics or economics courses, and anyone who passed calculus once but wants to actually understand it this time.
The language used throughout is plain English. Mathematical notation is introduced carefully and always explained when it first appears. Every worked example shows every line: not just the "interesting" ones. The goal is that a student who has never seen a derivative before can start at the First Principles page and work through to the Integration page and arrive with a genuine, connected understanding of the entire subject: not just a bag of formulas that will dissolve the moment the exam pressure lifts.
§ 05A Final Word
Maths is not magic. It is not talent. It is not a gift that some people are born with and others are not. It is a skill: and every skill is learnable.
If you are struggling with calculus right now, it does not mean you cannot do maths. It means you have not yet encountered the explanation that makes it click for you. That explanation exists. Sometimes it takes ten different approaches before the right one lands. This site is one approach, built with the explicit aim of being the explanation that finally makes it click.
Keep going. The moment of clarity, when it comes, is worth every minute of confusion that preceded it. And it always comes, if you keep asking why.