Derivatives / Differential Calculus: Definitions, Rules

polynomial extreme values — The derivative is another name for the slope of a tangent line at a point. Here, the derivatives at points A and B are zero.

Contents:

The Basics

What is a Derivative?
- Formula: Limit Definition of a Derivative
TI-89 Examples
Differentiation Notation

Specific Functions

More Definitions & Examples

What is a Derivative?

Simply put, it’s the instantaneous rate of change. It tells you how quickly the relationship between your input (x) and output (y) is changing at any exact point in time.

The following formula gives a more precise (i.e. more mathematical) definition.

Limit Definition of a Derivative

derivative definition

There are short cuts, but when you first start learning calculus you’ll be using the formula.

It’s not uncommon to get to the end of a semester and find that you still really don’t know exactly what one is! That’s because the definition isn’t immediately intuitive; you really get to grasp what one is after you’ve practiced—and practiced. It’s like knowing what an embouchure is in clarinet playing; you can be told that it’s a tongue placement, but it takes many weeks (sometimes months) of practice before you really get a good grasp of the perfect enbouchere and why it’s important.

You can find derivatives in a few different ways. Finding derivatives using the limit definition of a derivative is one way, but it does require some strong algebra skills. Watch the video for a couple of quick step-by-step examples:

Limit definition of derivative examples

Watch this video on YouTube.

Example problem #1: Find the derivative of f(x) = √(4x + 1)

Step 1: Insert the function into the formula. The function is √(4x + 1), so:
f'(x) = lim _{Δx → 0} √( 4( x + Δx ) + 1 – √(4x + 1) ) / Δx.
If this looks confusing, all we’ve done is changed “x” in the formula to x + Δx in the first part of the formula.

Step 2: Use algebra to work the formula. Here’s where you’ll benefit from strong algebra skills, because every formula is different.

Multiply the top and the bottom by √( 4( x + Δx) + 1 + √(4x + 1):
f'(x) = lim _{Δx → 0} √( 4( x + Δx) + 1 – √(4x + 1) ) * √( 4( x + Δx) + 1 + √(4x + 1) / Δx* √( 4( x + Δx) + 1 + √(4x + 1)
which reduces to:
= lim _{Δx → 0} 4(x + Δx) + 1 – (4x + 1) / Δx(√ (4x + Δx) + 1) + √ 4x + 1
Distribute the 4:
= lim _{Δx → 0} (4x + 4Δx + 1 – 4x – 1) / (Δx(√ (4x + Δx) + 1) + √ (4x + 1)
Delete terms. In this case you can delete 4x, Δx and 1.
= lim _{Δx → 0} 4 / ((√ (4x + Δx) + 1) + √ 4x + 1))

Step 3:Take the limit. The Δx will drop out (because it’s an insignificant increment). Again, strong algebra skills will help here:
= 4 / ((√ (4x + 1) + √ 4x + 1)
= 4 / 2 √(4x + 1)
= 2 / √(4x + 1)

That’s it!

Why is The Derivative Important?

Very basically, they are important because they allow you to extract information you didn’t know was there. For example, if you know where an object is (i.e. you have a position function), you can use the derivative to find velocity, acceleration, or jerk (rate of change of acceleration). How? The derivative of…

…position is velocity.
…velocity is acceleration.
…acceleration is jerk.

You can keep on taking derivatives (e.g. fourth, fifth), extracting more and more information from that simple position function. And it doesn’t just work with position; Calculus can work with any function.

TI-89 Derivative Examples

Finding derivatives on the TI 89 or the TI 89 Titanium involves the same steps. That’s because the two calculators are essentially the same, except for a few bells and whistles on the Titanium, like extra memory. These upgrades don’t affect how you find derivatives.

Derivative TI 89/Titanium Steps
Evaluate a derivative at a specific value,
Finding higher derivatives (2nd, 3rd…),

TI 89/Titanium Steps

Example problem: Find the derivative of f(x) = 3x.

Step 1: Press F3.

Step 2: Select “1: d( differentiate”. Use the down arrow key or type “1” to select it.

Step 3: Press ENTER.

Step 4: Type in a function, followed by a comma. For example, if your function is 3x then type “3x,”.

Step 5: Type X.

Step 6: Type a closing parenthesis symbol.

Step 7: Press ENTER. The solution (3 in this example) is on the right of the screen.

2. Evaluate a Derivative at a Specific Value

Step 1: Follow Steps 1 through 4 above.

Press The F3 button
Select “1: d( differentiate”
Press ENTER
Type your function name, followed by a comma.
Type X

Step 2: Close the parentheses “)”, then type a vertical bar (called the “with” symbol). On the TI-89, you’ll find the “|” key on the left hand side. Don’t press enter yet.

Step 3: Type the value you’re trying to find. For example, type x=3 if you’re trying to find the value of a derivative at x = 3. Press ENTER.

3. Finding Higher Derivatives (2nd, 3rd…)

Example problem: Find the second derivative of f(x) = 3x² on the TI 89.

Step 1: Follow Steps 1 through 4 in the first section above:

Press The F3 button
Select “1: d( differentiate”
Press ENTER
Type your function name (3x²), followed by a comma.
Type X

Step 2: Type a comma, then the number of the derivative you’re trying to find. For this example, you want to find the second derivative, so type “2”. You should have the following syntax:

d(3x^2,x)|x=2

Step 3: Press ENTER.

That’s it! You’re done!

Derivative Tables (Quick Reference Guide to Common Derivatives

Calculus Tables: Contents

Derivatives:

1. General Rules

2. Powers and Polynomials

3. Trigonometric Functions

4. Exponential Functions and Logarithmic Functions

5. Inverse Trigonometric Functions

6. Hyperbolic and Inverse Hyperbolic Functions

See: What is a Hyperbolic Function?

References

Wikibooks Tables of Derivatives

Fractional Calculus and the Half Derivative

Watch this 10 minute introduction, or read on below:

Fractional Calculus in 10 minutes.

Watch this video on YouTube.

Fractional calculus is when you extend the definition of an nth order derivative (e.g. first derivative, second derivative,…) by allowing n to have a fractional value.

Back in 1695, Leibniz (founder of modern Calculus) received a letter from mathematician L’Hopital, asking about what would happen if the “n” in Dⁿx/Dxⁿ was 1/2. Leibniz’s response: “It will lead to a paradox, from which one-day useful consequences will be drawn.”

There is no obvious graphical understanding of fractional calculus, and the basic rules we have derived for classical calculus do not all apply—or become very complicated. Simple ideas like the basis for the chain rule or product rule are no longer simple ideas when it comes to this new kind of calculus, and sometimes they just don’t work.

Simple Example of Fractional Calculus: the Derivative of a Power Function

Calculating the derivative of a power function is one of the simplest tasks in calculus, so it is perhaps a good place to begin exploring how a half derivative function might behave.

Let’s define our function of interest, f(x), as:

Our knowledge of calculus tells us that the first derivative will be

And we can repeat this a times to get the generalized result
$derivative formula for fractional calculus$
Now we know that the factorial is equivalent to the gamma function,

We can substitute that in in the general formula above, which leaves us with
$fractional calculus example$

Now, to find the half derivative, we have to substitute k = 1 and a = ½.
$fractional calculus half derivative$

In the graph below, this half derivative is graphed in purple, alongside the original function (in blue) and the first derivative (in red).

$Graph of first and half derivatives in fractional calculus$

Directional Derivative

The directional derivative tells you the instantaneous rate of change of a function in a particular direction.

You can write this type of derivative as:

That notation specifies you are looking at the rate of change for the function f(x,y,z) at a specific point (x₀, y₀, z₀). The symbol ∇ is called “nabla” or “del“.

This idea is actually a generalization of the idea of a partial derivative. For a partial derivative, you take the rate of change along one of the coordinate curves while holding all other coordinates constant. For a directional derivative, you must take into account all parts of your directional vector.

Directional Derivative of a Scalar Function

The directional derivative of a scalar function (i.e. a one dimensional function) is relatively easy to define. Along a vector v, it is given by:

This represents the rate of change of the function f in the direction of the vector v with respect to time, right at the point x.

Properties of the Directional Derivative

There’s one particularly good thing about the directional derivative; many of the properties of the ordinary derivatives hold for it as well.

For example, if our functions f and g are differentiable at a point p:

The sum rule holds:
For any constant c, the constant factor rule holds:
The product rule (otherwise known as Liebniz’s rule) holds:
And, if g is differentiable at p and h is differentiable at g(p), the chain rule also holds:

What is Automatic Differentiation?

Automatic differentiation (autodiff) uses a computer to calculate derivatives at some specified value, using a mechanical application of the chain rule.

It doesn’t give you a formula, but rather the value of the derivative at the point of interest.

Applications for Automatic Differentiation

Autodiff is used in:

Data assimilation,
Design optimization,
Inverse problems,
Numerical methods,
Sensitivity analysis.

It can deal with very complex functions; One of the largest applications was a 1.6 million line code (written in Fortran 77) for research in fluid dynamics.

Advantages

Advantages include:

It’s efficient and stable,
Answers are fairly precise,
It’s generally a good choice if you need to calculate the derivative at a point.
It’s generally considered a better choice than other computer-based differentiation like finite differencing, symbolic differentiation, or hand coding (Gebremedhin, 2014).

Types of Automatic Differentiation

There are two main types of autodiff:

Forward mode autodiff,
Reverse mode autodiff.

Forward mode (also known as director tangent linear mode), is usually used for calculating directional derivatives. It involves finding derivatives of intermediate variables with respect to independent variables. Propagating from one statement to the other is done via the chain rule.

Reverse mode (also known as backward, adjoint or cotangent linear mode) computes the derivatives of dependent variables with respect to intermediate values. Again, propagating from one statement to the next is done according to the chain rule.

These two modes are mathematically equivalent and based on the same principles, but they may require different amounts of time and computer memory.

Automatic Differentiation: References

Bischof, Bucker, Rasch, Slusanschi, and Lang. Automatic Differentiation of the General-Purpose Computational Fluid Dynamics Package. Journal of Fluids Engineering, 2007, volume 129 no 5. pages 652—658. Abstract retrieved from http://fluidsengineering.asmedigitalcollection.asme.org/article.aspx?articleid=1431286 on March 31, 2019Gebremedhin, A. (2014). Research, Automatic Differentiation. Retrieved June 1, 2019 from: https://www.cs.purdue.edu/homes/agebreme/research/coloring-derivatives.html
Bücker, Schiller, Hovland et al. Autodiff. Retrieved from http://www.autodiff.org/?module=Introduction on March 31, 2019.
Grosse, Roger. CSC32 Lecture 10: Automatic Differentiation. The University of Toronto CS. Retrieved from https://www.cs.toronto.edu/~rgrosse/courses/csc321_2018/slides/lec10.pdf on March 30, 2019.
Kourounis, D. et al. (2017). Compile-Time Symbolic Differentiation Using C++Expression Templates. Retrieved June 1, 2019 from: http://arxiv-export-lb.library.cornell.edu/pdf/1705.01729
Wang, Chi-Feng. Automatic Differentiation, Explained. How Do Computers Calculate Derivatives? Retrieved from https://towardsdatascience.com/automatic-differentiation-explained-b4ba8e60c2ad on March 31, 2019.

Epiderivatives

Epiderivatives are derivatives given by epigraphs (the set of points lying on or above a function’s graph) in set-valued optimization problems.

The method unifies the delta and nabla approaches to calculus of variations on time scales (Girejko, et al., 2010).

Epiderivative Definition

Epiderivatives generally fall into the categories of:

Contingent epiderivatives (introduced by Jahn & Rauh in 1997) or
Generalized contingent epiderivatives.

The set-valued contingent derivative of a set-valued function H at a certain point is the map with a graph equaling the Bouligand contingent cone of the graph of H at the chosen point (Bigi & Castellani, 2002). More formally, it can be described using notation:

A contingent epiderivative of F at (x, y) is defined as (Rodriguez-Marin et al., 1997):

“…A single-valued map DF (x, y): X → Y whose epigraph coincides with the contingent cone to the epigraph of F at (x, y) i.e. epi (DF (x, y) = T (epi(F), (x, y).”

The generalized contingent epiderivative of F at (x, y) is defined as the following set-valued map:
D_g F (x, y): X → 2^Y if
D_g F (x, y) (x) = Min ({y ∈ Y: (x, y) ∈ T (epi(F), (x, y))}).

Higher Order Derivatives

Higher order derivatives are any derivative other than the first (Second, third, fourth, …). The derivative of a function is also a function, so you can keep on taking derivatives until your function becomes f(x) = 0 (at which point, it isn’t possible to take the derivative any more).

Taking the derivative over and over again might seem like a pedantic exercise, but higher order derivatives have many uses, especially in physics and engineering.

Example of Finding Higher Order Derivatives

The first derivative of the function f(x) = x⁴ – 5x² + 12x – 13 is:
f′(x) = 4x³ – 10x + 12 (found using the power rule).

But you can differentiate that function again. As you’re differentiating two times, it’s called the second derivative. Using the power rule again, you get: f′′(x) = 12x² – 10

You can keep on taking the derivative of this particular function five times, when the fifth derivative equals zero. You can’t take a derivative of zero, so you’ll stop there.

Higher Derivatives of the Position Function

Higher order derivatives have many theoretical uses, but they also have a few practical ones. Exactly what kind of information you extract depends upon what kind of function you start with. For example, let’s say you start with a position function.

The first derivative of the position function gives you the velocity function, which gives you the velocity of the object.

The second derivative (of the position function) gives you the object’s acceleration.

The third derivative gives you the jerk— the rate of change of acceleration. It’s called a “jerk” because that’s how a quick change in acceleration feels. Imagine being on a descending elevator that slows down suddenly: that jerking sensation you feel is due to the change acceleration.

The fourth derivative of the position function gives the rate of change of the “jerk”. At this point, the higher order derivatives become more theoretical, but they do have some important uses, especially in safety of high velocity objects (like roller coasters!).

What is the Third Derivative?

The third derivative is the derivative of the second derivative. In other words, it’s the rate of change or slope of the curve of the second derivative.

Jerks and Lurches

The third derivative of the position function is called a jerk, which is the rate of change of acceleration. Let’s suppose that s(t) is an object’s position function:

The first derivative, s′(t), is the object’s velocity function,
The second derivative, s′′(t), is its acceleration,
s′′′(t) is the object’s jerk.

It’s called a jerk (or, less commonly, jolt, lurch, or surge) because changes in acceleration tend to feel “jerky”, especially large ones. A smooth elevator ride feels exactly like that— smooth. But ride on the Tower of Terror, an accelerated drop tower in Disney World’s Hollywood Studios, your stomach will tell you why changes in acceleration are also called “lurches”.

Notation

For any function f(x), f′′′(x) can be defined by several different notations, all of which mean the same thing (from Stewart, 2009):

Example

General steps:

Take the derivative of the function (using established derivative rules). This is called the first derivative.
Take the derivative of the new function (i.e. the first derivative). This new function is called the second derivative.
Take the derivative a third time.

Example : What is the third derivative of f (x) = xⁿ?
Solution, using the power rule multiple times:

f′(x) = nx^{n − 1}
f′′′(x) = n(x^{n − 1})′ = n(n − 1) x^{n − 2}
f′′′(x) = n(n − 1)(x^{n − 2})′ = n(n − 1)(n − 2) x^{n − 3}

Fourth and Higher Derivatives

Fourth and higher derivatives are less common. In order to find the fourth derivative, take the derivative another time (i.e. take the derivative of the 3rd derivative). Essentially, you can keep on going on and on to infinity with taking derivatives—it is possible to find the hundredth or thousandth or millionth derivative. However, in reality (and with the types of equations you are likely to encounter), you’ll likely only be able to take derivatives up to the fifth derivative. After that, you’ll probably end up with a constant—and while second, third, and fourth derivatives can give you useful information about a function’s behavior, the hundredth derivative does not.

Sixth Derivative

The sixth derivative (also called pop or pounce) is the result of taking the derivative of a function (usually, the position function) six times. In other words, it’s the derivative of the fifth derivative.

Higher order derivatives, like this one, are rarely seen outside of physics. And when they do occur, they are not usually of much importance. In physics, you would usually find an approximation using a Taylor series, rather than go through the laborious process of finding sixth derivatives. For simple functions, like the one below, finding the sixth derivative is relatively easy. But most real world problems are going to involve functions of greater complexity, which means you’re going to want to approximate the answer with a Taylor series anyway.

Sixth Derivative: Example Problem

Example question: What is the sixth derivative of f(x) = x⁶ – 3x⁴ + 9 x – 11?

Solution: Use the power rule and constant rule to take the derivatives six times:

f′(x) = 6x⁵ – 12x³ + 9 (First derivative)
f′′(x) = 30x⁴ – 36x² (Second derivative)
f′′′(x) = 120x³ – 72x (Third derivative)
f⁽⁴⁾ = 360x² – 72 (Fourth derivative)
f⁽⁵⁾ = 720x (Fifth derivative)
f⁽⁶⁾ = 720 (Sixth derivative)

Snap, Crackle, and Pop

The sixth derivative is called pop after the Snap, Crackle, and Pop of Rice Krispies. J. Codner et al came up with those names (see footnote 17 in Scott et al.) in response to a question asked on the USENET sci.physics newsgroup.

Fifth Derivative (Crackle)

References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.
Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999
Beyer, W. H. “Derivatives.” CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 229-232, 1987.
Bigi, G. & Castellani, M. (2002). K-epiderivatives for set-valued functions and optimization. Mathematical Methods of Operations Research. 401-412.
Girejko, et al., The contingent epiderivative and the calculus of variations on time scales. in Optimization—A Journal of Mathematical Programming and Operations Research. 2010.
Griewank, A. Principles and Techniques of Algorithmic Differentiation. Philadelphia, PA: SIAM, 2000.
Jahn, J. Rauh, R. Contingent epiderivatives and set valued optimization
Math. Methods Oper. Res., 46 (1997), pp. 193-211
Khan, A. et al. (2015). Set-valued Optimization: An Introduction with Applications (Vector Optimization). Springer.
Kimeu, Joseph M., “Fractional Calculus: Definitions and Applications” (2009). Masters Theses & Specialist Projects. Paper 115. digitalcommons.wku.edu/theses/115 Retrieved from https://digitalcommons.wku.edu/cgi/viewcontent.cgi?article=1115&context=theses on April 12, 2018
Kisak, P. (Ed.) (2017). An Overview of Jerk Physics: ” The Meaning of The 3rd Derivative “.
ScienceDirect Fractional Derivatives & Calculus. Retrieved from https://www.sciencedirect.com/topics/physics-and-astronomy/fractional-calculus on April 8, 2019
Rodriguez-Marin, L. & Sama, M. About Contingent Epiderivatives. J. Math. Anal. Appl. 327 (2007). 745-762.
Scott, J. Some Simple Chaotic Jerk Functions in Am. J. Phys., Vol. 65, No. 6, June.
Stewart, J. (2009). Calculus: Concepts and Contexts. Cengage Learning.
Stewart, James. Calculus: Early Transcendentals. Partial Derivatives: Directional Derivs and the Gradient Vector. Retrieved from https://math.libretexts.org/Bookshelves/Calculus/Map%3A_Calculus_-_Early_Transcendentals_(Stewart)/14%3A_Partial_Derivatives/14.6%3A_Directional_Derivatives_and_the_Gradient_Vector on July 27, 2019.
TI89 Users Manual.
Troltzsch , F. Optimal Control and Differential Equations. Maik Nauka/Interperiodica Publishing, 1995.
Viljoen, C & Van der Merwe, L. (1999). Elementary Statistics: Calculations & Interest for Business & Economics. Pearson South Africa.

The Basics

Specific Functions

More Definitions & Examples

Why is The Derivative Important?

2. Evaluate a Derivative at a Specific Value

3. Finding Higher Derivatives (2nd, 3rd…)

4. Exponential Functions and Logarithmic Functions

References

Simple Example of Fractional Calculus: the Derivative of a Power Function

Directional Derivative of a Scalar Function

Properties of the Directional Derivative

Applications for Automatic Differentiation

Advantages

Types of Automatic Differentiation

Automatic Differentiation: References

Epiderivative Definition

Example of Finding Higher Order Derivatives

Higher Derivatives of the Position Function

Jerks and Lurches

Notation

Example

Fourth and Higher Derivatives

Sixth Derivative: Example Problem

Snap, Crackle, and Pop

Related Articles

References