Differential Calculus

Differential Calculus Equation – sciencephotolibrary


Differential calculus is the study of rates of change and slopes of curves. df/dt is the derivative (d) and represents the slope of a tangent at any single point on a curve. On a graph t represents the horizontal axis and f represents the vertical axis. The process of calculating the derivative is known as differentiation. Rates of change values are applied in many everyday applications, for example fuel consumption of a car, velocity of an object and yield of a crop.1


I still remember my first day of calc 1. It was a summer evening around 6:30pm. Professor Burger walked into the class and said: “good evening guys my name is professor Burger and I’m your calc instructor. Let me start the class by telling you a secret, if you fail calculus is not because of the calculus, it is because you can not do algebra”

Through my school years I have seen this story repeating over and over. People don’t struggle with the calculus, people struggle with the algebra.7

Calculus is too often taught as Johnny-Carsonish-stupid-pet-tricks for symbolic integration. No universal insight, just special, specific tricks. I took college calculus from a book that made the inane statement “Integration is more difficult than differentiation because integration is an inverse operation.” Bullshit! One can just as well define integration as the primitive operation (just the limit of a different process) and differentiation as the inverse operation.

Integration is more difficult because, unlike differentiation, you don’t have an immediate easy product rule breaking down products into factors you can readily integrate. (Integration by parts is not a simple, direct product rule.)

To grasp calculus you must truly understand the definition of a limit and its application in the definition of derivatives and integration. (Not to worry, both Newton and Leibniz also had a hard time with this.) When you can take the standard definition of a limit, swap epsilon and delta, replace “no matter how small” with “no mater how large” and that no longer gives you a worry, you understand the definition. (I do not suggest using this on a test unless you want to risk testing the teacher…)

Now for what they hide in calculus class: In the real world almost no one does arithmetic by hand anymore except for trivial problems. Likewise, very few do algebra or calculus by hand anymore for any but trivial problems. Symbolic algebra, differentiation, and integration programs are used. Most ‘real” problems don’t have closed form solutions – they are solved numerically.8

All of the other answers so far blame the student or the instruction. While there may be some merit in these, the main issue is that calculus is fundamentally different from prior topics.

I quote from my answer to Why is calculus considered disturbing and confusing by most students?:

I’m going to lean a bit on some mathematical learning theory. I think what is challenging about the traditional calculus course is how quickly functions, limits, and then derivatives need to be reified — turned from processes or procedures into mathematical objects.

At the time they enter calculus, I doubt most students even have a very objectified view of functions. Mostly for them, a function is (at best) still a rule that tells you what output you get from a given input. Right out of the gate in calculus, though, we pretty much require the function to be viewed as an object itself.

Likewise with limits. Students naturally view “limits” as some process of “getting close to”, but we ask them very quickly to reify that understanding into limit-as-object (speaking of “the limit” rather than “taking the limit”, for instance). We rarely even recognize the need for this reification, much less provide sufficient time and practice for it.

And again with the derivative. We first define the derivative as a limit (which itself students still view, at best, as a process), then quickly switch over to a bunch of procedural rules for calculating a derivative. Then, usually in the very next section, we pivot and expect students to have an objectified view of the derivative as a function itself. Again, we as instructors rarely recognize this cognitive quantum leap, and therefore rarely provide students opportunity to make it.9


In mathematics, differential calculus is used,10

  • To find the rate of change of a quantity with respect to other
  • In case of finding a function is increasing or decreasing functions in a graph
  • To find the maximum and minimum value of a curve
  • To find the approximate value of small change in a quantity

Real-life applications of differential calculus are:

  • Calculation of profit and loss with respect to business using graphs
  • Calculation of the rate of change of the temperature
  • Calculation of speed or distance covered such as miles per hour, kilometres per hour, etc.,
  • To derive many Physics equations


See Theoretical Knowledge Vs Practical Application.


When solving problems about differential calculus, I can think of a few tips that you may find useful:6

  1. Know the definitions. For example, make sure you know what it means for a function to be increasing or decreasing on an interval. Make sure you know how a limit is defined, and what it means for a function to be continuous at a point. Make sure you know what a rate of change is is and how to express it as a derivative. Knowing very well the definitions you’re working with is essential not just for differential calculus but for all of mathematics.
  2. Be methodical. Until you have enough practice that you can start skipping steps, you need to be very careful with every operation. Make sure you’re not doing too many things from one line to the next. If you are, leave some of them for the next line. This is also helpful for any kind of mathematics, not just differential calculus.
  3. Keep a table of derivatives handy. And consult it often. You don’t want to write the derivative of cosxcos⁡x as sinxsin⁡x when it is actually −sinx−sin⁡x.
  4. Keep another table with the differentiation rules. Make sure you can identify a product of functions (for the product rule), a quotient of functions (for the quotient rule) and a composition of functions (for the chain rule).
  5. Always check your results. When you’re done, go back and make sure your answer is correct. Especially if you have applied problems, make sure your answer makes sense in the context of the problem. You don’t want to build a fence that’s −50−50 feet long. This is also helpful in any other field of mathematics besides differential calculus.

Many of the References and Additional Reading websites, and Videos will assist you with solving differentials.

As some professors say: “It is intuitively obvious to even the most casual observer.”


I do not normally provide examples, but in calculus there are many formulas (e.g., trigonometric formulas) and rules (e.g., the chain rule) that can be used when solving a differential. Also, I would like to show you the usefulness of the differential.

Example 1 – Differential Solution

highermaths provides us a good example of solving a differential as shown in the image below.

Find the solution of the differential

Since I am relearning calculus, I was stumped by Steps 1 and 4. Here is what I learned:

  • Step 1: We need to differentiate both sides of ( x – y ) = t wrt (with respect to) x. Why? To simplify (reduce) the equation to a simpler form so we can solve for the slope (Differentiation of y with respect to x). So ( x – y ) becomes Differentiation of x wrt x minus y with respect to x and t becomes Differentiation of t with respect to x.
  • Step 2: In this step we substitute into our original equation t for ( x – y ) so cos( x – y ) becomes cos(t), and also solve for Differentiation of t with respect to x from Step 1.
  • Step 4: Here is where trigonometry comes in! We need to recognize and use the half angle formula.
    HAF 1
    HAF 2
    HAF 3
    HAF 4
    HAF 5
  • Step 6: Remember, in trigonometry, that Relationship between the sin and csc (cosecant)

QED! Another example that the steps to solve the differential are intuitively obvious to even the most casual observer.

From the wikiHow website:5

Write out complete solutions, even for practice and homework. Your teacher will usually require you to show your work—or write out every step—when you’re taking a math test. Get in the habit of doing that every time you work a problem, even if it’s just something you’re doing to practice.

  • In addition to helping you on exams, showing your work makes it easier to go back and see where you made a mistake if you get the wrong answer.
  • For instance, if you’re solving “2x = 3+1,” don’t just skip to “x = 2.” Write out “2x = 4,” then “2x/2 = 4/2,” then “x = 2.”

A trough is being filled up with swill. The trough is 10 feet long, and its cross-section is an isosceles triangle with a base of 2 feet and a height of 2 feet 6 inches (with the vertex at the bottom, of course). Swill’s being poured in at a rate of 5 cubic feet per minute. When the depth of the swill is 1 foot 3 inches, how fast is the swill level rising? 2 3

Filling up a trough – SchoolBag

Rather than show you the solution, please visit the Your Problems Are Solved: Differentiation to the Rescue – Differentiation – Calculus For Dummies page. There are other examples on this page that demonstrate the power of differentiation.


1 LIBRARY, SCIENCE. 2021. “Differential Calculus Equation – Stock Image – C020/0564”. Science Photo Library. https://www.sciencephoto.com/media/594565/view/differential-calculus-equation.

2 “Your Problems Are Solved: Differentiation To The Rescue – Differentiation – Calculus For Dummies”. 2021. Educational Materials. https://schoolbag.info/mathematics/calculus_1/13.html.

3 ⭐ Ryan, Mark. Calculus for Dummies. Hoboken, NJ: Wiley Publishing, Inc., 2003.

4 “Limits In Calculus (Definition, Properties And Examples)”. 2021. BYJUS. https://byjus.com/maths/limits/.

5 “14 Ways To Study Math – wikiHow”. 2021. wikihow.com. https://www.wikihow.com/Study-Math.

6 García, Fabio. “What Are The Best Ways To Solve Differential Calculus?” 2022. Quora. https://www.quora.com/What-are-the-best-ways-to-solve-differential-calculus.

7 Liza, Martin E. “Why Is Calculus Hard For Some Students?”. 2022. Quora. https://www.quora.com/Why-is-calculus-hard-for-some-students.

8 Mclane, Chas. “Why Is Calculus Hard For Some Students?”. 2022. Quora. https://www.quora.com/Why-is-calculus-hard-for-some-students.

9 Wasburn-Moses, Jered. “Why Is Calculus Hard For Some Students?” 2022. Quora. https://www.quora.com/Why-is-calculus-hard-for-some-students.

10 ⭐ “Differential Calculus (Formulas And Examples)”. 2022. BYJUS. https://byjus.com/maths/differential-calculus/.

Additional Reading

“Derivative Of Sin X – Formula, Proof, Graph | Differentiation Of Sin X”. 2021. cuemath. https://www.cuemath.com/calculus/derivative-of-sin-x/.

“Differentiation Rules – Finding The Derivative Of A Difference Of Functions”. 2013. sk19math.blogspot.com. https://sk19math.blogspot.com/2013/05/derivative-difference-functions.html.

“Differentiation Rules – Finding The Derivative Of A Sum Of Functions”. 2013. sk19math.blogspot.com. https://sk19math.blogspot.com/2013/05/derivative-sum-functions.html.

“How To Understand Derivatives (Calculus)”. 2021. Medium. https://mikebeneschan.medium.com/how-to-understand-derivatives-calculus-ad21bb847533.

⭐ Strogatz, Steven. Infinite powers: how calculus reveals the secrets of the universe. Boston: Houghton Mifflin Harcourt: 2019.

⭐ “7 Derivative Rules You Should Know”. 2021. Medium. https://albertming88.medium.com/7-derivative-rules-you-should-know-ce405b8f9f4d.

Important concepts and formulas for the basis of differential calculus

Rule #1: Constant Function Rule

If is a function that is a constant c, then the derivative of f is zero.

Rule #2: Constant Multiple Rule

The derivative of a constant multiplied by a function is equal to the constant itself multiplied by the derivative of the function.

Rule #3: Power Rule

The derivative of x raised to a power is the value of the power times x raised to the quantity of the power minus one.

Rule #4: Sum Rule

The derivative of a sum is the sum of the derivatives.

Rule #5: Product Rule

The derivative of a product of two functions is the sum of the first function multiplied by the derivative of the second function and the derivative of the first function multiplied by the second function.

Rule #6: Quotient Rule

The derivative of the quotient of two functions is the difference of the function in the denominator multiplied by the derivative of the function in the numerator and the function in the numerator multiplied by the derivative of the function in the denominator, all divided by the function in the denominator squared (this may be confusing, so refer at the equation in the article). This rule may be the most complicated out of the bunch. Unfortunately, there are no tricks involved here, and the process of using the quotient rule is about as boring as it looks.

Rule #7: Chain Rule

The derivative of a composition of two functions is the derivative of the “outer function” with the input of the “inside function” multiplied by the derivative of the “inside function.”

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

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