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Now, since we understand a couple of basic things about differentiation, let's start seeing how to do it.

Rule #1: Any constant's derivative is equal to 0.

A constant is a definite quantity like 5, or 45 or maybe 567; so for example:



Its derivative (with respect to x), is:



You can change the constant to any value you wish; its derivative is still 0. And when in an equation you are told that a certain symbol is to be treated like a constant, then its derivative will be 0. For example:



And you are told that c is a constant by definition. Then the function will have derivative 0, since c is considered to be a constant (of unknown quantity).



More examples I will do more at the end of this post.

Rule #2: if you have a function like [math]f(x)=x^n, then its derivative will be [math]f'(x) =nx^{n-1}

Let's take the parabola of my first post:



Notice that n=2; so n-1 = 1, and the resulting derivative will be:



One should always remember that [math]x^1=x; there is no need to put the power of 1 to any variable or constant. Also we should remember that [math]x^0=1 no matter what the variable, quantity is.

So, if we have:



then n=1, so n-1 = 0, then the derivative of the above is [math]f'(x)=1.

Let's do a couple more examples:



So, we have n=19, and n-1 = 19-1 = 18, so its derivative will be [math]f'(x)=19x^18. How about the second derivative though? The rule is that when you have a constant before a power that constant remains unaffected in differentiating. When I will explain more complex functions like [math]x\times sin(x)or [math]x(sin(x^2)), you will see why this constant is not differentiated to 0. For the moment, just take it as part of the definition.

The definition then is amended to:



Its derivative will be:



So, if we want to make the second derivative of the above what happens?



We have a=19, n=18, n-1=17. So plugging in the numbers we have:




Another very important thing to remember is the following: if a function is made up of addition of other functions, then its derivative will be the addition of the derivatives of those functions.

In mathematical terms:



then the derivative will be: