Differentiation is a calculus function involving a set of rules, which mathematicians use to find the rate of change. These rules are extremely simple, and once you learn how to use them, you will be able to do them in your head and not require a calculator. This article is for beginners, year 12, kindergarten, and for dummies.


Differentiating Equation

Here is the general rule showing how to differentiate any power of x. Beginners often call the power rule “differentiating equation”, but it is just a set of steps. Let us say that n is any number, then all you have to do is to multiply x by n, and then subtract 1 from the power, as shown by the animation above.

Here is a brain training graphic you could use. Just stare at it for a while, and imagine n is any number. When n-1 appears, subtract 1 from the number. It might take a little time to understand it so just stare at the animation until you get it. Once you become an expert mathematician, you will be able to do this as second nature.

x² Example Animation

Let us start with the basics by differentiating x² (aka x^2). The result is 2x as you can see in the animation above.

Multiply by the power, and then subtract 1 from the power. If you had x³ (aka x^3) then that would become 3x², because you multiply by the power and then subtract 1 from the power.


Although some of the modern and vintage calculators are able to manage differentiation, it is better to be able to do this without any aid. If you are planning on a professional career then you should learn to do this by hand and build some intuition.


Sum Rule

Sum Rule

Difference Rule

Calculus History

I always remind myself that calculus was invented hundreds of years before even Newton and Leibniz came across it. Man in ancient India knew of the infinite series, which is the core concept of calculus. Whilst the zero was conceptualised in the 3rd century AD, exponents and logarithms were known in the first millennium and earlier.

Interestingly, although European historians would like to give Newton the credit for inventing Calculus, Newton himself did not take any credit for it. He claimed only to have made it more rigorous.

This Article Continues...

Differentiation Chain Rule
Differentiation Quotient Rule
Differentiation Product Rule
Differentiation Formulas
Differentiation of ln x
Differentiation of Exponential
Differentiation of tan x
Differentiation of log x
Differentiation x^x
Differentiation y=a^x
Differentiation from First Principles