Geometric and physical interpretation of derivative

Now that you understand what a derivative is, you might be curious about its significance or its practical uses. In this topic, you'll learn about its graphical interpretation and its implications in diverse fields like physics and biology; consider this as an introduction to its application in optimization!

This understanding will transform the derivative from a seemingly abstract concept that is tricky to calculate into a fundamental tool for discerning the properties of a function. Furthermore, you'll gain the capability to evaluate the instantaneous change of a variable determined by another, such as the expansion of a population over time.

The Tangent Line

Consider a function $f:\mathbb{R} \to \mathbb{R}$ and a point $a$ in its domain. The graph of function $f$ forms a curve in the plane. Your task is to identify the tangent line to the graph of $f$ at the point $P =\left(a, \, f(a) \right)$.

For any other adjacent point $x$ that is not equal to $a$, you can pinpoint the point $Q =\left(x, \, f(x) \right)$. The slope of the secant line that connects $P$ with $Q$ equals $\frac{f(x) - f(a)}{x-a}$:

Now, when $x$ comes close to $a$, watch the slopes:

Did you see it? The slope of the secant line from $P$ to $Q$ seems to approach the slope of the tangent line at $P$. As $x$ comes close to $a$, the slopes almost align with the value known as the derivative of $f$ at $a$:

$f'(a) = \lim_{x \rightarrow a} \frac{f(x) - f(a)}{x-a}$

Since a line is determined by its slope and any of its points, the derivative provides the equation of a tangent line.

Understanding the Curve

You might not find computing tangent lines to be the most exciting activity, huh? Nevertheless, the slopes of these lines can reveal quite a bit about your function. Begin by examining the function below along with some of its tangent lines. Try to identify a pattern in their slopes:

Did you notice what happens when the slopes are positive? The function increases near those points! Similarly, when the slopes are negative, the function decreases near those points. To sum up, the sign of the derivative dictates the behavior of $f$ around a point.

But there's more to infer from the image. At points where the slopes steepen, the function changes more rapidly. In simpler terms, the derivative's size (in absolute value) lets you know how quickly the function is increasing or decreasing.

Now, pay attention to what happens to the slopes near $P$ and $Q$, the points where the function reaches a minimum and a maximum, respectively. The tangents start to level off, meaning their slopes approach $0$. In essence, the extreme points have a derivative equal to $0$. This is a crucial application of derivatives, which we'll consider later in the topic.

Physics

Imagine an object moving in a straight line. You can represent its position, $s$, as a function $f$ of time, $t$. In this case, $s = f(t)$ illustrates the movement of the object:

Here, within the time interval $[a, x]$, the average velocity is determined by comparing the displacement $f(x) - f(a)$ with the elapsed time $x - a$:

$$\frac{f(x) - f(a)}{x-a}$$

Seems familiar, right? It's identical to the slope of the secant line from $\left(a, \, f(a) \right)$ to $\left(x, \, f(x) \right)$. You can calculate the velocity for shorter time intervals just like before by letting $x$ draw closer to $a$. The derivative of displacement as a function of time is thus defined as the instantaneous velocity, $v$:

$v(a) = f'(a)$If you consider velocity as a function of time, then its derivative at $a$ is its instantaneous rate of change. As you may guess, this is referred to as acceleration. But let's dive a bit deeper from here.

Rates of change

Imagine a quantity that's influenced by another through a function $f$. When the independent quantity shifts from $a$ to $x$, it prompts the dependent one to change from $f(a)$ to $f(x)$. Consequently, the difference quotient $\frac{f(x) - f(a)}{x-a}$is recognized as the rate of change in the interval $[a, x]$. Building on your prior steps, you can think of the average rate of change over increasingly smaller intervals by letting $x$ approach $a$. The limit of these average rates of change, also known as the derivative, becomes the instantaneous rate of change.

Quite surprisingly, instantaneous rates of change hold vital importance in numerous fields such as:

• Biology: If the population of a bacteria colony is modeled as a time-dependent function, then the derivative provides insight into the rate of the population's growth.

• Economics: In a scenario where the cost a company incurs in producing a product is viewed as a function of the number of such products, then the instantaneous rate of change is interpreted as the marginal cost.

• Electricity: When electric charges move through a surface, they generate a current. To define it more formally, the current at a specific time is the instantaneous rate of change of the charge.

Now let's discuss machine learning. The primary task in supervised learning is to fine-tune model parameters to minimize a loss function, which quantifies the difference between model predictions and actual outputs. As you've already established, the derivative of this loss function, respect to the model parameters, points out the direction and intensity of the adjustments needed to minimize the loss.

The most commonly employed method for this optimization is gradient descent. With the aid of the derivative, the model parameters are updated iteratively. But how does that happen? It's done by measuring the value of the derivative: the smaller its value, the closer we get to the minimum point. Therefore, always keep in mind that the derivative plays a central role in the learning process of neural networks by minimizing the errors they produce!

Conclusion

• The line tangent to the graph of $f$ at point $P =\left(a, \, f(a) \right)$ runs through $P$ with a slope of $f'(a)$.

• If $f'(x) > 0$ on an interval, $f$ increases in that interval.

• If $f'(x) < 0$ on an interval, $f$ decreases in that interval.

• If $f$ reaches a maximum or minimum at $a$, then $f'(a)=0$.

• The derivative of displacement with respect to time defines the instantaneous velocity.

• In essence, the derivative $f'(a)$ denotes the instantaneous rate at which $f$ changes at $a$.