Higher Order Derivatives : Let's begin this section with the given function.
f ( x ) = 5x3 - 3x2 + 10 x - 5
By this point we have to be able to differentiate this function without any problems. Doing this we obtain,
f ′ ( x ) = 15x2 - 6 x + 10
Now, it is a function and thus it can be differentiated. Following is the notation that we'll utilize for that, as well as the derivative.
f ′′ ( x ) = ( f ′ ( x ))′ = 30x - 6
This is called the second derivative and f ′ (x) is called the first derivative.
Again, thus it is a function we can differentiate it again. It will be called the third derivative. Following is that derivative in addition to the notation for the third derivative.
f ′′′ ( x ) = ( f ′′ ( x ))′ = 30
Continuing, we can differentiate again. It is called, oddly sufficient, the fourth derivative. We're also going to be altering notation at this point. We can keep adding on primes, however that will get cumbersome after awhile.
f ( 4) ( x ) = ( f ′′′ ( x ))′ = 0
This procedure can continue however notice that we will acquire zero for all derivatives after this point. These derivatives lead us to the given fact regarding the differentiation of polynomials.
Fact : If p(x) refer for a polynomial of degree n (that means the largest exponent in the polynomial) then,
P( k ) ( x ) = 0 for k ≥ n + 1
We will have to be careful along with the "non-prime" notation for derivatives. Assume each of the following.
f (2) ( x ) = f ′′ ( x )
f 2 (x ) = [ f ( x )]2
In the exponent the presence of parenthesis indicates differentiation whereas the absence of parenthesis denotes exponentiation.
Collectively the second, third, fourth, etc. derivatives are called as higher order derivatives.
Let's take a look at couple of examples of higher order derivatives.