Okay, since we know what derivatives mean.
We can develope a few rules that will help us.
1)Derivative of Constant Function: The derivative of a function (some number) is zero, that is, . One way is is to think of this as the slope of a vertical line (which is zero). Another way is through the limit:
2)Derivative of a Sum: You might have expected
where are some functions. Again, this is easy to show through the limit.
Let and .
3)Derivative of a Difference: The same thing as addition. That is,
Note: If you want to sound cool and impress your teachers you can say rules #1,2,3 are true because "Differenciation is a linear transformation for the vector space of differenciable functions over the field of reals". Or can you can "Differenciation is a homomorphism". Basically "derivative" has the property that the derivative of a sum is the sum of the derivatives. You will found many operators during you math learning that have this propert, pay attention to them.
4)Derivative of a Product: The rule says,
The derivation is a bit strange, but it relys on a trick mathematicians love to use.
Again using the same convention for as above we can write,
Add and subtract (thus no change in the expression).
Thus, some factoring,
Note the only questionable step was,
Because, in the other post I spoke about limits and I mentioned you can only substitute the value when the function is continous at the point. It happens to be true. Because a differenciable function at a point is continous at a point, this is extremely useful in theoretical demonstrations.
5)Derivative of a Quoteint: If then,
This time we add and subtract
6)The Power Rule: The power rule says that for any in positive integer we have,
I am going to present two proofs.
One the standard way.
The second which is my way.
The standard way is by the binomial expansion. The only important thing to know is that,
Where, are some numbers which we really do not care about.
From this we subtract the orginal function .
Then we divide through by ,
But everything to the right of is zero because a exists.
Here is my way....
Note, by product rule,
Thus, by product rule again,
In fact this pattern hold for some positive integer of functions,
Thus, the derivative by Hacker's product rule above is,
The easy way to remember this rule is to bring down the exponent in front of the varaible and reduce exponent by 1.
The nice thing is that though we proved it only for positive integers it hold for any power.
Thus, we can find,
Time for examples.
But I am sure you do not need them, you are good with algebra.
Example 5: .
By, the sum and power rule we have,
By the product rule,
But the power and sum rules,
We could have also multiplied out,
By, power and sum rules,
Example 7: .
We can write,
Though it is not a positive integer the rule still holds,
Now we reach the most important rule. The "chain rule".
I will state it, and explain why it is called chain rule.
First we need to know what a composite of functions is. If we have a functions and . The composition of these functions is a new function,
where is the "inner" and is the "outer".
Note, , (always)meaning it is not commutative.
Example 8: If and .
7)Limit of a Composition: The rule says,
Take the derivative of the inside, multiply it by the composition of the derivative of outer function.
There is an easy to do it.
Then, the composition,
As if we can cancel the 's.
We can write,
We can write,
But we can write more,
, , .
Hence the name "chain rule".
I will not prove the chain rule. The proof is difficult. But there is a nice trick, a weaker result, that makes it an easy prove. I do not want to post it because that will be the next problem of the week.
Thus, now you know the most important rule about derivatives!
Find for the following functions:
*6)Prove the chain rule for polynomial functions.
That is given,
Find their composition and then compute the derivative.
And compute the derivative via chain rule.
And then show the results match.