Differentiation VI: Higher Order Derivatives

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Lesson Summary: 
We motivate the fact that we can find as many other derivatives of a function as possible and emphasize the use of the Leibniz notation to describe them. We calculate all of the derivatives of f(x)=x2 and f(x)=ex. We provide links to lessons that detail the meaning of the second derivative.
Lesson Inputs: 
Differentiation I: An Overview of Differential Calculus
Differentiation II: The Gory Details of Calculating Derivatives
Leibniz Notation
Lesson Outputs: 
Geometry of Functions VIII: The Second Derivative Test
Geometry of Functions X: Determining Concavity From Second Derivatives

SAMPLE LESSON

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Lesson: 

Given any function f(x), we can attempt to find its derivative f′(x). Since f′(x) is a function can't we try to find its derivative? Absolutely! If we were to do this and it existed, then we would denote it by <Sign in to see all the formulas>. Now let's keep going. Since <Sign in to see all the formulas> is a function can't we try to find its derivative? Absolutely! If we were to do this and it existed, then we would denote it by <Sign in to see all the formulas>. Since <Sign in to see all the formulas> is a function can't we try to find its derivative? Absolutely! . . .

I think you get the point now. We can always try to find as many derivatives of a function as we can. We refer to these derivatives as higher order derivatives. More specifically, we refer to <Sign in to see all the formulas> as the second derivative, <Sign in to see all the formulas> as the third derivative, etc. Our prime notation can get very cumbersome so we will typically start to use the Leibniz notation instead of the prime notation for higher order derivatives. For example, a 15th derivative (if it existed) would be denoted by

<Sign in to see all the formulas>


and not by <Sign in to see all the formulas>. Typically you will see the prime notation for the first or second derivatives but almost never for the third derivative and above. At that point the Leibniz notation is used to denote the n-th order derivative by

<Sign in to see all the formulas>

As an example let's find all of the derivatives of <Sign in to see all the formulas>. Well we know that <Sign in to see all the formulas>. We know from our differentiation formulas that

<Sign in to see all the formulas>


Since 2 is a constant its derivative is zero. So let's write out the derivatives we have found:

Derivative Notation Formula
First

<Sign in to see all the formulas>

<Sign in to see all the formulas>

Second

<Sign in to see all the formulas>

<Sign in to see all the formulas>

Third

<Sign in to see all the formulas>

<Sign in to see all the formulas>


Notice that since zero is a constant and its derivative is zero, we can actually find an infinite number of derivatives. However, all of them are zero above the second derivative. We could write this more formally as

<Sign in to see all the formulas>


for <Sign in to see all the formulas>. Let's re-write our results to reflect this:

Derivative Notation Formula
First

<Sign in to see all the formulas>

<Sign in to see all the formulas>

Second

<Sign in to see all the formulas>

<Sign in to see all the formulas>

Third and Above

<Sign in to see all the formulas>

<Sign in to see all the formulas>


Note that the symbol <Sign in to see all the formulas> means "for all".

Let's do one more important example. Let's find the derivatives of <Sign in to see all the formulas>. Recall that <Sign in to see all the formulas>. In other words, for the exponential function its derivative is itself.

Derivative Notation Formula
First

<Sign in to see all the formulas>

<Sign in to see all the formulas>

Second

<Sign in to see all the formulas>

<Sign in to see all the formulas>

Third

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

n-th Derivative

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>

<Sign in to see all the formulas>


In other words, we have an infinite number of non-zero derivatives for the exponential function.

Finally, while we have motivated the fact that we can mechanically and repetitively keep finding more and more derivatives of a function, we have not discussed what they mean. The most important thing to understand is the meaning of the second derivative. It explains everyday concepts of how fast something is increasing or decreasing and geometrically describes the concavity of a function.