Calculus Using Algebra: Part 2

Before you read this: make sure to read part 1 of this series first to clear up some basics. Now let's get started!

As we found out last time, we used algebra to find the derivative of X^2. But what about X^3? Or X^4? Or, in general, what about X^N? Let's find out!

Higher Power Derivatives

Last time, we expanded the polynomial (A+E)^2 and found the coefficient of Epsilon, removing the higher order term of E^2 to get 2A as our derivative. But wait, the only difference between this and the derivative of X^3 is the power we raise it to! So if we expand the polynomial (A+E)^N, we get the derivative of X^N! Using Pascal's Triangle to find the polynomial expansion's Epsilon coefficient (which will be NA^(N-1), see Pascal's Triangle: A wonder of mathematics for why), we find that the derivative of X^N is N*A^(N-1)! This means, for example, the Derivative of X^3 is 3x^2!

Note: For fractional coefficient polynomials like 1/3x^3, you can just take the derivative of X^3 and multiply it by 1/3. For Multi-term polynomials, we can take the derivative of each term and add them.

Integrals

The Integral is the opposite of the derivative, meaning you can rephrase the question "What is the integral of BLANK" as "What number has a derivative of BLANK?" Let's start by finding the integral of the simple X^2!

The integral of a second-degree polynomial is always a third degree polynomial (An N degree polynomial's integral will be an N+1 degree polynomial.) Let's call the Integral AX^3. The derivative of AX^3 is 3A X^2. 3A X^2= X^2. 3A= 1, A=1/3, so our integral is 1/3X^3+C! (You must add +C to the end of an Integral as the Constant of Integration. I know. It's weird.

Like and I'll post part 3: Generalizing Integrals!