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 at least understand what the derivatives are so first let's start with the trig functions if I wanted 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 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 was going to be negative sine of X so derivative of sine is cosine derivative cosine is negative sine and then finally the derivative the derivative of tangent of X tangent of X is equal to is equal to 1 over cosine squared of X which is equal to the secant squared secant squared of X once again these are all very good things to know now let's talk a little bit about Exponential's and logarithms so the derivative and actually this is one of the coolest was one of the coolest results and once again speaks to how cool he is as a number the derivative with respect to X of e to the X we use a drumroll 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 draw a graph e to the X so that's my y-axis let's say that this right over here is my x-axis x-axis so if I have very negative values of X either to a very negative value we're approaching 0 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 and start 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 the 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 what's this with a tangent line here turns out that is also one amazing if I go to X is equal to X is equal to one right over here the function the function evaluated here gets us e to the one 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 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 got 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