Exponential functions differentiation
Derivatives of sin(x), cos(x), tan(x), eˣ & ln(x)
Let's get some exposure to the derivatives of some of the most common functions. We're not going to prove them in this video, but at least understand what the derivatives are. So first, let's start with the trig functions. If I want to take the derivative with respect to x of sine of x, this is going to be equal to cosine of x. And if you look at their graphs, it'll make intuitive sense. Once again I have not proved it here, but this is a good thing to know, that the derivative of sine of x is cosine of x. Now what about the derivative of cosine of x? What about the derivative with respect to x of cosine of x? Well, this one's going to be negative sine of x. So the derivative of sine is cosine, and the derivative cosine is negative sine. And then finally, the derivative of tangent of x is equal to 1 over cosine squared of x, which is equal to the secant squared of x. Once again, these are all very good things to know. Now let's talk a little bit about exponentials and logarithms. So the derivative-- and actually, this is one of the coolest results, and it once again speaks to how cool e is as a number, the derivative with respect to x of e to the x-- we need a drum roll for this one. This is one of the coolest things in mathematics. The derivative of e to the x is e to the x. Now what does that tell us? And I have to take a little pause here, because this is just so exciting. So let me graph e to the x. So that's my y-axis. Let's say that this right over here is my x-axis. So if I have very negative values of x, e to a very negative value, we are approaching zero. And then e to the 0 is 1, so that's going to be 1 right over there. So it's going to look something like this. And then it's an exponential. It's going to go, it's going to start increasing really, really, really, really, really fast. So let's say that's the graph of y is equal to e to the x. What this tells us is that at any point-- so let's say I go right over here. I say when x is equal to 0, e to the 0 is 1, what's the slope of the tangent line here? Turns out that is also 1. Amazing. If I go to x is equal to 1 right over here, the function evaluated here gets us e to the 1 power or just e. And what's the slope of the tangent line right over here? It is also e. At any point right over here, the slope of the tangent line is equal to the value of the function at that point. This is amazing. This is what is so cool about e. Anyway, that's not the point of this video. This video is to give you a catalog of all of the derivatives that you might really need. So then finally, if we're thinking about the derivative with respect to x of the natural log of x, this is going to be equal to-- and this is also fascinating. This is equal to 1 over x or x to the negative 1. So somehow, we have our natural log has kind of inserted itself into-- when you take the derivative, as filling in the gap that the power rule left vacant, which is, is there some function whose derivative is equal to x to the negative 1? The power rule gave us functions whose derivatives might be x to the negative 2, x to the negative 3, or x to the squared or x to the fifth. But it left the x to the negative 1 vacant, and it's filled by the natural log of x. Now I haven't proved it here. I've just catalogued these for you. And then we can use these in future videos, and we'll prove them in future videos.