So, there I was during IRC, just chatting along, when someone asked some random calculus question, I got it right, and then they asked me to explain how to do derivatives the “easy” way. I was originally going to do this through MSN Messenger with the nifty little drawing conversation feature, but alas, the person in question didn’t support Handwriting. Instead, I’m now going to make a short little blog post about how one can easily do a derivative. This is just the easy stuff, remember, I’m not getting into derivatives of fractional functions or something like that.

Another thing, these graphical examples were done quickly in paint, so don’t make fun of me ;_;

First, we have a simple function f(x) = x^2 – 3x – 4, as displayed in the picture below:

For simple derivatives such as these, we first have to remember some properties of exponents, such as: any variable by itself (or with a coefficient) is *variable*^1 . If a coefficient is by itself, we can say that it the coefficient multiplied times the variable to the zeroeth power — *variable*^0 . We can use these properties to modify our original equation to look something like this:

Now that we have the modifications out of the way, we can get to the actual derivation. For equations like these, to get the derivative, you multiply the power by the coefficient, and then subtract one from the power, as shown in the drawing:

Then, you use a bit of simplification, and you’ll come out with f**‘**(x) = 2x – 2, as shown below:

See? It’s not really that hard. :)

Sexy, I bet I’ll be learning this shiz in a week or two :'(

I can still remember our teacher showing us this after she taught us the really long way!

You have to learn the long way (limits) first because thats what derivatives are. If someone did not understand what a limit was and you showed them the easy way to get the derivative then it would be even more confusing.

It’s probably more useful to just remember that the derivative of a constant is zero. There’s really no need to throw an imaginary x^0 on the 4 just to illustrate the power rule.

@Clifford: Yes, you’re correct. The person in question that wanted me to explain how to do the “easy” way already knew about limits and how to derive a function that way, they just wanted to know how to do it the other way, as well.

@leatherdaddy: Yes, that’s true, but I wanted to make sure that they saw where the rule that a derivative of a constant is zero comes from.

This was helpful. I missed like two weeks of pre calc, came back in time for the product, chain and quotiant rule but had no idea how to do a simple derivative. This has really caught me up to speed. Thankums’

I like how everyone I talk to that is in Precalc is doing stuff like this… Whereas, I, who goes to a pricey private school, am currently doing really in depth Trig stuff. Granted, my math teacher was a cancer ridden, senile genius until he was replaced with a creepy Nam’ veteran who is also a magician. I’m not making that up by the way… Oh well, guess I’ll just have to stick to my commitment that I’ll teach myself some calculus this summer.

yessss. it makes sense

Awesomely simple. I wish textbooks and or teachers would speak in such simple language. Especially in a course designed for people who only need the most basic knowledge of calculus

thank you so much :)

being in first year university this just hit me across the head like a smelly fish (stupid high school didn’t teach this..)

my gosh… you make it easy :)

This one is easy