Introduction

Hello fellow readers! Today, FiveHive presents the first subtopic in Unit 4 in AP Calculus. In this article, we will discuss the actual impact and significance of derivatives in different contexts. 

A Deeper Understanding of Derivatives

At the most fundamental level, the derivative is an instantaneous rate of change of the output variable with respect to the input variable. 

Up until this point, we have been working with derivatives from a computational standpoint in Units 2 and 3, learning ideas like Chain Rule, Implicit Differentiation, etc. Those lessons established derivatives as the slope of a tangent line for a function. This is consistent with our original statement, and is a graphical representation of the original phase of “instantaneous rate of change.” Slopes represent rates of change, and tangent lines are lines that touch one singular point. Taking the slope of that means you are taking the rate of change (slope) at one particular instant. The instantaneous rate of change measures how $y$ (output variable) changes with respect to $x$ (input variable). 

We will extend this understanding to real-world applications and different contexts. This is because you can replace the notion of $x$ and $y$ with any real-world unit. Note that $f(x)$ and $y$ are used interchangeably, since those variables are all notation for the output variable. For instance, the derivative of a profit function based on goods sold would reveal information about the profit gained per unit of goods sold. Expressed generically:

$f’(x)=\frac{\text{Unit of }f}{\text{Unit of }x}$

Another way to interpret this is by understanding that since $f’(x)$ outputs a slope, and the core definition of a slope is $\frac{\Delta y}{\Delta x}$, then it naturally follows that the units of derivative $f’(x)$ is the unit for $f$ divided by the unit for $x$. 

Helpful Tip: Using the above idea is an example of dimensional analysis, which involves checking the units with each quantity. This is extremely helpful in interpreting derivatives correctly. AP loves defining functions as the rate of change of $y$ with respect to $x$, and then asking what the derivative of that function is. The answer would be the rate of change of rate of change of $y$ with respect to $x$.

Example:Water is being pumped into the pool. The rate that water is being pumped into the pool is modeled by a differentiable function $f$, where $f(t)$ is measured in gallons per second and $t$ is measured in seconds since pumping began. What are the correct units of $f’(t)$.

Solution:

We know that $f’(t)=\frac{\text{Unit of }f}{\text{Unit of }t}$. Therefore, $f’(t)$ has the unit of $\frac{\frac{\text{gallons}}{\text{second}}}{\text{second}}$. This can be read aloud as “gallons per second per second” or as gallons per second squared (gallons per second${}^2$).

Practice