Basic derivative rules (Part 1) | Derivative rules | AP Calculus AB | Khan Academy

So these are both ways that you will see limit-based definitions of derivatives. Usually, this is if you're thinking about the derivative at a point. Here, if you're thinking about the derivative in general, but these are both equivalent. They're both based on the slope of a tangent line or the instantaneous rate of change. Using these, I want to establish some of the core properties of derivatives for us.

The first one that I'm going to do will seem like common sense, or maybe it will once we talk about it a little bit. So, if F of x, if our function is equal to a constant value, well then F prime of x is going to be equal to zero. Now, why does that make intuitive sense? Well, we could graph it. We could graph it. So, if that's my y-axis, that's my x-axis. If I wanted to graph y = F of x, it's going to look like that, where this is at the value y is equal to K.

So this is y is equal to F of x. Notice, no matter what you change x, y does not change. The slope of the tangent line here, well frankly, is the same line. It has a slope of zero. No matter how y is just not changing here, we could use either of these definitions to establish that even further, establish it using these limit definitions.

So let's see the limit, and as h approaches zero of f of x + h. Well, no matter what we input into our function, we get K. So f of x plus h would be K minus F of x. Well, no matter what we put into that function, we get K over h. Well, this is just going to be 0 over h, so this limit is just going to be equal to zero.

So, f prime of x for any x, the derivative is zero. And you see that here, that this slope of the tangent line for any x is equal to zero. So, if someone walks up to you on the street and says, "Okay, h of x, h of x, h of x is equal to pi, what is h prime of x?" You say, "Well, pi, that's just a constant value. The value of our function is not changing as we change our x. The slope of the tangent line there, the instantaneous rate of change, is going to be equal to zero."