In this lecture, we'll introduce one of the most fundamental concepts in calculus, namely the derivative function of a given function. By the end of

this lecture, we'll not only define the derivative function but also demonstrate how to compute it for simple polynomial

functions. So let's jump right in. What is the definition of the derivative function, often denoted as f prime of x or

df/dx, for a given function f(x)? It is defined as the limit of this expression inside the square brackets,

called the difference quotient, as h approaches zero. To understand this expression better, let's consider this graph of a

general function f(x). On this graph, we label two points: point P with coordinates x and f(x), and

point Q with coordinates x + h and f(x + h). Now we'll draw a secant line

that connects these two points, P and Q. Why do we care about this secant line? Because its slope is going to give us

crucial insight. As we know, the slope of a line is defined as the ratio of the change in y

to the change in x. Then, using the coordinates of P and Q, the change in y, or delta y, is f(x + h) - f(x), and

the change in x, or delta x, is x + h - x, or simply h. Then the slope of the secant line becomes

f(x + h) - f(x) / h, and this is exactly the expression for the difference quotient in the above

definition. In other words, the difference quotient represents the slope of the secant line connecting the points P and Q. Finally, as the definition states, we now

need to take the limit of this difference quotient as h approaches zero. Let's visualize what happens

during this process. Given the secant line shown here in gray, connecting the two points P and Q on the curve f(x), and as h becomes

smaller and smaller, the two points on the curve, P and Q, move closer together, as you can see in this

animation. The secant line begins to converge to the line tangent to the curve at X. This means, as H approaches zero, the

slopes of the secant line must also converge to the slope of the tangent line. So, in a sense, the derivative, f prime

of x, represents the slope of the tangent line to the function f at the point x. And this is the core idea of a

derivative: as the limit of the slopes of secant lines. So feel free to pause the video and take a moment to understand

this very important concept. Now, how can we apply this definition to compute the derivative of a simple polynomial function? Let's work

through an example together. Consider the following second-degree polynomial function, f of x, given by x².

We'll now use the definition of the derivative to show that this derivative function is 2x. First, if f of x is x²,

then f of x + h is simply (x + h)². Expanding this expression using the distributive property, or FOILing, we get

f of x + h = x² + 2xh + h². Again, by definition, f prime of x is the

limit as h approaches zero of this difference quotient. Substituting f(x + h) with x² + 2xh + h² and f(x) with x², we get the

following expression. Now, after canceling x² from the numerator, this becomes (2xh + h²) /

h. Let's note that here both terms in the numerator can be divided by h. Then this simplifies further to 2x + h.

Finally, as h approaches zero, the h term vanishes, leaving us with f prime of x = 2x. And this is exactly what we wanted to

show. And it wasn't too bad, was it? I hope this video helped you understand one of the most fundamental

definitions in calculus, namely the definition of the derivative function. So if you enjoy this lecture, be sure to like and subscribe to my channel for

Calculus 1: The derivative function (Definition and Example)

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