# Understanding Calculus: A Simplified Approach to Derivatives

## Finding derivatives in calculus has many practical uses, and can be an invaluable tool.

The derivative in calculus is an important concept. It provides a rate of change of a function, not over long periods, but at an instance. The simple way of stating this is the derivative of a function is the slope of the tangent line to the function at a point. For most functions the derivative function is a known form, and we can use a table to find the derivative function which is usually simply called the derivative.

There are several notations for the derivative in common use. Here we will use df(x)/dx.

A simple example is to think of a company with sales (in thousands of units) per year given be f(x) = x4, where x is the number of years. How are sales projected to be changing at the end of year 2?

The derivative of f(x) = xn , df(x)/dx is nxn-1, so dx4/dx = 4x3. This means the function f(x) = x4 has a tangent line with a slope of 4x3 after x years. So the rate of change of sales after two years is df(2)/dx = 4(2)3 or 32, meaning sales are projected to be increasing at a rate of thirty-two thousand units per year.

From the above example it is obvious the derivative has many everyday uses, and because of this the derivative is extremely valuable. Another use is velocity is the rat, which is the of change of the displacement, and acceleration is the rate of change of velocity.

### Where Do Derivative Formulas Come from?

The formulas come from theory.  The idea of a slope of a tangent line starts with chords that have endpoints close enough to each other that they approximate a segment of a tangent line.

The process is easy to understand, and understanding is essential to acceptance.  To follow an explanation and an example through the process I recommend watching the video below.

### Does Combining Derivatives Follow Arithmetic Rules for Combining Functions?

If two functions are combined with addition or subtraction, even if either or both have constant multipliers, the derivative can be found by finding the individual derivatives and combining them in the same way.  This means d(af(x) ± bg(x))/dx = a(df(x)/dx) ± b(dg(x)/dx).

The derivatives of the product of two functions, the quotient of two functions, and the composition of functions have special ways of being handled.  These are handled as shown in the videos below.

To find my other videos easily I recommend subscribing to my YouTube channel.

### What Else Can We Do with Derivatives?

Derivatives can help find where functions are at their maximum or minimum, or even local maxima or minima, This can be aided by finding where the tangent is zero or is undefined.  That is an involved discussion and is reserved for another article.

### Derivatives in Calculus

 How to destroy Calculus - Limits and Derivatives: Third book in Col...View on Amazon 27 Calculus Math Flash Cards - Common Derivatives and IntegralsView on Amazon

### What is implicit Differentiation

Occasionally, when a derivative is needed for a function that cannot be isolated, we use the chain rule and include the derivative of the function as dy(x)/dx.  Using y(x) in implicit differentiation is common.  Solving in terms of both x and y is allowed.

Often, when x appears in an exponent, we use logarithms.  When we do that, we take the derivative of ln y.  Use of implicit differentiation is important in such cases.

### Using L’Hopital’s Rule

##### (Also Spelled L'Hospital's Rule

When taking limits, it sometimes occurs that we have indeterminant form, such as zero divided by zero.  If cancellation does not resolve the problem, we can take the derivatives of the numerator and of the denominator, and try to obtain that limit.  In such cases, the replacing of the expressions in a limit with their derivatives may produce a limit.  The video below contains the details.

### Keep Watching for More!

I am planning to add other articles with more videos embedded.  These will be forthcoming.

Updated: 08/05/2024, blackspanielgallery
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blackspanielgallery 2 hours ago

It depends on the high school. In this state one needs 12 college hours in a subject to be allowed to teach it, provided one has the other requirements. Some schools teach calculus and college level algebra, and more, but many high school teachers do not have the background to teach advanced math. The 12 hours can, and often do, include courses that did not count for math majors. In high school I had, albeit a long time ago, two algebra courses, one trig and one geometry course. The algebra was not at the level of college algebra.

So fewer people write at calculus and other college level courses.

DerdriuMarriner 5 hours ago

Your answer seven comment boxes down advises us that "I started in calculus because there are so many videos on YouTube by high school teachers in lower math, but I am adding an article in algebra today."

What subjects are taught by high-school teachers of "lower math"?

blackspanielgallery 21 days ago

That was standard, and still is, at the university I attended. Now more students take a four course sequence, 3, 3, 4, and 3 hours, splitting the same work into four parts. The university found 5 hour calculus courses were too intense for many.

blackspanielgallery 21 days ago

We had algebra, geometry, trig, then another algebra.

DerdriuMarriner 22 days ago

The computer crashed before I commenced another component of my comment-question below.

Your comment two boxes down describes three calculus courses completed during your college years.

Did your not doing high-school calculus demand your so doing that number, that type as college coursework?

DerdriuMarriner 22 days ago

Thank you for your comment below in answer to my previous observation and question.

Your answer 4 comment boxes down advises us that in "high school I did not get calculus."

Did your school, unlike mine, not have any calculus course (so much fun going around with a slide rule!) or did you opt not to take an available, extant course?

blackspanielgallery 22 days ago

In college I had three calculus courses, 2 for 5 credit hours each and one for 3 semester hours.
Many advanced college courses are theoretical, meaning if one is good in geometry one should be good in advanced math since the skill set of proving things is similar.

blackspanielgallery 22 days ago

In college I had three calculus courses, 2 for 5 credit hours each and one for 3 semester hours.
Many advanced college courses are theoretical, meaning if one is good in geometry one should be good in advanced math since the skill set of proving things is similar.

DerdriuMarriner 23 days ago

Thank you for your comment below in answer to my previous observation and question.

My math courses all are from junior high school and senior high school. Geometry counted as my least favorite math class even as I found everything before and everything after it equally fascinating, practical and revelatory.

Would a Master of Science in mathematics therefore preclude having to take a course entirely devoted to calculus?

(It would be fascinating and useful to have such a degree that works algebra, geometry, trigonometry, algebra IV and calculus into a unified whole!)

blackspanielgallery 23 days ago

I high school I did not get calculus. However, having a Master of Science in mathematics more than filled in my calculus, but it was in physics, my other field in which I have two degrees, a BS and a MS in applied physics that I really learned the value of calculus and the techniques became clearer. I started in calculus because there are so many videos on YouTube by high school teachers in lower math, but I am adding an article in algebra today. I probably go a little deeper than many others.

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