Differentiation I: An Overview of Differential Calculus

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Lesson Summary: 
Differential calculus is the study of the slopes of functions and their application to real world problems. The slope of a function $f(x)$ at a specific point is called a derivative and is denoted by two common notations: $f'(x)$ and $\frac{df}{dx}$. We provide a listing of real world examples where derivatives are used to describe real world phenomenon.
Lesson Inputs: 
The Slope Of A Straight Line
An Overview of Calculus
Lesson Outputs: 
Differentiation V: Derivatives and Rates of Change
The Formula of the Tangent Line

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Lesson: 

Differential calculus is the study of the slopes of functions and their application to real world problems. Recall that the slope of a straight line is its "rise" over its "run":

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More importantly, we can determine the slope anywhere along the straight line and always get the same answer.

However, for any other function, the slope can be different depending upon where we look. For example, the picture below shows that the function <Sign in to see all the formulas> has a slope of -10 at $x=-5$, 0 at $x=0$, and 4 at $x=2$. So depending on where we are at on the x-axis, our slope will be different from point to point.


In general, functions can have many slopes depending on where you look.

There are typically two common notations for the slope of a function. The slope of a function at a point $x$ of a function $f(x)$ can be denoted by

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and is typically called the "derivative" of the function or more formally " the derivative of 'f' with respect to 'x'."

So why is the slope of a function so important that we have an entire branch of mathematics for it? Because there are many real world phenomenon that are described by the slope of a function. Many things in life depend upon something else. In those situations we are also wanting to know how one variable changes with respect to another. Here are some examples:

Derivatives In The Real World
  • The population of a country can fluctuate over time. Two of the obvious inputs affecting the population is the death rate and the birth rate. If we denote the population, birth, and deaths at a point in time as $P(t)$, $B(t)$, and $D(t)$, respectively, then the birth and death rates are <Sign in to see all the formulas> and <Sign in to see all the formulas>, respectively. Furthermore, the change in the population over some small time interval is <Sign in to see all the formulas>. In fact, it seems that the population will increase or decrease based on whether there are more or less births and deaths. One could even say that <Sign in to see all the formulas>. In plain English, we say that the rate of increase (or decrease) of the population depends on changes in birth rates and death rates.
  • Bacterial growth is the process by which a bacterium will divide into two cells ("daughters") identical to the "parent" bacterium. This exponential growth phase is characterized by a constant rate of doubling of the bacteria population as well as the rate of population growth during each consecutive time period. The population as a function of time can be described by <Sign in to see all the formulas> where $\mu$ is the growth rate that measures the number of cell divisions per time. Eventually, nutrients used to fuel cell division will be depleted resulting in a slower growth rate (smaller value of $\mu$).
  • One of the key parameters in studying flu epidemics is the basic reproductive number $R_0$. $R_0$ is the number of new, secondary infections created from a single primary infection case. $R_0$ is important to know for those fighting infectious diseases because it relays the risk of a new outbreak causing an epidemic. If <Sign in to see all the formulas> then the possibility of an epidemic is high. Furthermore, the final size of the epidemic can also be estimated from $R_0$. $R_0$ is dependent on the duration of the infection, number of contacts per time, and the number of infections passed on per contact. As anyone of these parameters increases, $R_0$ increases. If we use $D$, $N$, and $I$ to denote the duration of the infection, number of contacts per time, and the number of infections passed on per contact, respectively, then saying that $R_0$ increases with respect to these parameters means that

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  • Speed is a derivative. Let's say that you have traveled 60 miles from home in an hour. On average then you have traveled 60 miles per hour. But if at any time you can measure your distance from home and at what time then you can establish the ratio

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    where $D(t)$ is the distance at time $t$. The numerator is the distance traveled over a time $h$ divided by the time $h$. This is just the speed of the car between times $t$ and $t+h$. Taking the function limit as <Sign in to see all the formulas> gives the speed on the speedometer the instance you look at it. Just as speed is the derivative of distance with respect to time, acceleration is the derivative of speed with respect to time. With $D(t)$ as the distance, $S(t)$ as the speed, and $A(t)$ as the acceleration, then we have

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  • Typically air temperature gets cooler the higher in altitude you go. If we denote $T(A)$ as the temperature as a function of altitude then the air temperature getting cooler as the altitude gets higher can be denoted by the derivative

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    A temperature inversion layer is an area where the normal cooling of air temperature with increasing altitude is reversed and there is a layer of air that is warmer than the air below it. In this situation there are values of $A$ for which

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    Identifying temperature inversion layers is important to meteorologist. Temperature inversion layers block atmospheric flow and create areas of highly stable air. As a result of this, in areas with heavy pollution, inversion layers can trap pollutants creating unhealthy air and smog.
  • The number one cause of death is atherosclerotic heart disease which can cause heart attacks or sudden cardiac death. Research shows that the accumulation of calcium in coronary arteries, and not cholesterol, is a more accurate predictor of a future heart attack. This calcification is located near areas of advanced heart disease and autopsies show that there is a direct relation between the amount of calcification and the severity of the heart disease. Studies have shown that the probability of having at least one narrowed artery from heart disease increases with an increasing amount of calcium in the artieries. If we denote $H$ as the probability of having a heart attack and $C$ as the amount of calcium in the arteries, then we can summarize our knowledge by saying that <Sign in to see all the formulas>. In this example we want to know how our probability of having a heart attack increases with increased amounts of calcification in our arteries.
  • The value of U.S. Treasury and corporate bonds can be related to an interest rate called the yield-to-maturity. As the yield-to-maturity rises the value of the bonds decreases and when the yield-to-maturity decreases the value of the bonds increases. If we denote $V$ as the value of the bond and $y$ as the yield-to-maturity then this relationship between the value of the bond and the yield-to-maturity can be denoted by the derivative

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    The quantity $dV/dy$ is important as it measures the market risk of the bond value to changes in underlying interest rates. In fact, $dV/dy$ is the source for two major risk metrics used in valuing the risk of fixed income securities. The first is

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    which measure the change in value of the bond per one basis point change in the underlying interest rate. The second is bond duration. When using the explicit formula for the value-yield function, the quantity

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    is the weighted average lifetime of the bond's cashflows and therefore referred to as duration.

So now that we have a basic understanding of derivatives and what they are used for, here is how we actually find the derivative of a function.