Click here to purchase the entire book in PDF format. Derivation and Slope Back in Chapter 1.2 we looked at how to find the slope of a straight line, but what about if the line is a curve? Well, this gets us into a small problem because, if the line is curved, the its slope is different in different places. For example, take a look at the sinusoidal curve in Figure 1.18. Figure: A graph of $y = \sin(x)$ showing the point $x=2$ where we want to find the slope of the graph. Our goal is to find the slope at the point marked on the plot where $x=2$. In other words, we're looking for the slope of the tangent of the curve at $x=2$ as is shown in Figure 1.19. Figure: A graph of $y = \sin(x)$ showing the tangent to the curve at the point $x=2$. The slope of the tangent is the slope of the curve at that point. We could estimate the slope at this point by drawing a line through two points where $x_{1}=2$ and $x_{2}=3$ and then measuring the rise and run for that line as is shown in Figure 1.20. Figure: A graph of $y = \sin(x)$ showing an estimate to the tangent to the curve at the point $x=2$ by drawing a line through the curve at $x_{1}=2$ and $x_{2}=3$. As we can see in this example, the run for the line we've created is 1 (because it's $3 - 2$) and the rise is -0.768 (because it's $\sin(3) - \sin(2)$). Therefore the slope is -0.768. This method of approximating will give us a slope that is pretty close to the slope at the point we're interested in, but how to we make a better approximation? One way to do it is to reduce the distance between the point we're looking for and the points where we're drawing the line. For example, looking at Figure 1.21, we've changed the nearby points to $x_{1}=2$ and $x_{2}=2.5$, which gives us a line with a slope of -0.622. This is a little closer to the real answer, but still not perfect. Figure: A graph of $y = \sin(x)$ showing a better estimate to the tangent to the curve at the point $x=2$ by drawing a line through the curve at $x_{1}=2$ and $x_{2}=2.5$. As we get the points closer and closer together, the slope of the line gets closer and closer to the right answer. When the two points are infinitely close to $x=2$ (therefore, they are at $x_{1}=2$ and $x_{2}=2$ because two points that are infinitely close are in the same place) then the slope of the line is the slope of the curve at that point. What we're doing here is using the idea of a limit - as the run of the line (the horizontal distance between our two points) approaches the limit of 0, then the slope of the line approaches the slope of the curve at the point we're looking at. This is the idea behind something called the derivative of a function. The curve that we're looking at can be described by an equation where the value of $y$ is determined by some math that we do using the value of $x$. For example, if the equation is $$y = \sin(x)$$ (2.72) then we get the curve seen above in Figure 1.18. In this particular case, given a value of $x$, we can figure out the value of $y$. As a result we say that $y$ is a function of $x$ or $$y = f(x)$$ (2.73) So, the derivative of $f(x)$ is just another equation that gives you the slope of the curve at any value of $x$. In mathematical language, the derivative of $f(x)$ is written in one of two ways. This simplest is if we just we write $f'(x)$ which means ``the derivative (or the slope) of the function $f(x)$'' (remember: derivative is just a fancy word for slope). If you're dealing with an equation where $y = f(x)$ as we've seen above in this chapter, then you're looking for the ``derivative of $y$ with respect to $x$.'' This is just a fancier way of saying ``what's the slope of $f(x)$?'' We don't need to say ``$f$ of $x$'' because it's easier to say ``$y$'' but we need to say ``with respect to $x$'' because the slope changes as $x$ changes. Therefore there is a relationship between the derivative of $y$ and the value of $x$. If you want to use mathematical language to write ``derivative of $y$ with respect to $x$,'' you write $$\frac{dy}{dx}$$ (2.74) But, always remember, if $y = f(x)$ then $$\frac{dy}{dx} = f'(x)$$ (2.75) There is one important thing to remember when you see this symbol. $\frac{dy}{dx}$ is one thing that means ``the derivative of $y$ with respect to $x$.'' It is not something divided by something else. This is a stand-alone symbol that doesn't have anything to do with division. Let's look at a practical example. Any introductory calculus book will tell you that the derivative of a sine function is a cosine. (We don't need to ask why at this point, but if you think back to the spinning wheel and the horizontal and vertical components of its movement, then it might make sense intuitively.) What does this mean? Well, that means that the slope of a sine wave at some value of $x$ is equal to the cosine of the same value of $x$. This is written as is shown in Equation 1.76. $$f'(sin(x)) = cos(x)$$ (2.76) Just remember, if somebody walks up to you on the street and asks ``what's the derivative of the curve at $x=2$?'' what they're saying is ``what's the slope of the curve at $x=2$?'' If you want to calculate a derivative, then you can use the idea of a limit to do it. Think back a bit to when we were trying to find the tangent of a sine wave by plotting lines that crossed the sine wave at points closer and closer to the point we were interested in. Mathematically, what we were doing was finding the limit of the slope of the line, as the run between the two points approached 0. This is described in Equation 1.77 $$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$$ (2.77) Huh? What Equation 1.77 says is that we're looking for the value that the slope of the line drawn between two points separated in the x-axis by the value $h$ approaches as $h$ approaches 0. For example, in Figure 1.20, $h = 1$. In Figure 1.21, $h = 0.5$. Remember that $h$ is just the ``run'' and $f(x+h) - f(x)$ is just the rise of the triangle shown in those plots. As $h$ approaches 0, the slope of the hypotenuse gets closer and closer to the answer that we're looking for. Just to make things really miserable, I'll let you know that you can have beasts like the vicious double derivative written $f''(x)$. This just means that you're looking for the slope of the slope of a function. So, we've already seen that the derivative of a sine function is a cosine function (or $f'(sin(x)) = cos(x)$), therefore the double derivative of a sine function is the derivative of a cosine function (or $f''(sin(x)) = f'(cos(x))$). It just so happens that $f'(cos(x)) = -sin(x)$, therefore $f''(sin(x)) = -sin(x)$.