For this course, we only need a few simple derivatives. So let's run through those now. The first is the so-called product rule. If I have a, a function x to the n, where x is the variable and n is just any real number, the derivative of x to the n is just n times x to the n minus 1. So, I decrease the exponent by 1 and I multiply in front by n. the second one is that the derivative of constant is zero. So that's I think fairly intuitive if, if I have let's say I had a function f of x. So here's x, here's f of x. But that function happened to be a constant, so it's just unchanging. Doesn't matter what the value of x is it's always a constant. The slope of that line is always zero, it's a horizontal line. There's no change as a function of x. Now, another simple derivative is that, it's just an application of the power rule. The derivative of x to the first power is just 1, so I multiply by 1 in front, and x times x to the 0. And anything to the 0 power is unity, so this is 1. Now, we're going to need trigonometric functions, and the derivative of sine of x is cosine x, and the derivative of cosine x is minus sine of x. And we'll be talking more about this later and that's I think, if you start plotting sine of x and cosine x and look at the slope of the tangent lines, this will become apparent. Another one is the derivative of e to the x. Now, e is just a number, and it's a very special number. But it's only a number like pi, it's an irrational number. And it has the special property that if I take that number and raise it to the x power, then the slope of that function is always equal to the value of that function. And we'll talk more about that in a minute, and why that arises in certain types of problems. Now, we need two other simple rules for taking derivatives. The first one is the product rule. So, if I have two functions, f of x and g of x, and I multiply them together and I want to take the derivative of that product. The rule says, okay, it's the first function times the derivative of the second plus the second function times the derivative of the first. So, you just go in and take the derivatives one at a time, and leave the other function unchanged. So here's a simple example to illustrate that. If I have x times sine of x, so x is my f, sine of x is my g and I use this rule to take the derivative. Well, it's the first function x times the derivative of sine of x, which is cosine x plus the derivative of the first function. Well, the derivative of x is 1, and then g of x, unchanged is just sine of x. So here's that product. Another simple example is a to the a times x. So, the derivative of that, if I treat a as f, and x is g the derivative of that then is the first function a times the derivative of x, which is unity plus the derivative of the first function a, well that's just a constant, so that derivative is 0. So the whole second part of this disappears. So that's an application of the product rule. Now, the last rule we want to talk about is the so-called chain rule, and this is for taking derivatives of, of composite functions. So, if I have x as my variable and I form a function g of x, and then I take and I form a function of g of x, so it's a composite function. The derivative of that is I take the derivative of the outer most function trading g as the variable. And then I just mult, I go inside and then take the derivative of g itself and just multiply those in a, in chain fashion. So here's a simple illustration of that. I have sine of x to the second power. Well, that's just cosine of x to the second power, so that's just treating x squared as the variable and taking the derivative of sine of some variable. Well, that's just cosine of that variable. And then I go inside and take the derivative of x squared, which is just 2x. Another illustration of the chain rule is if I take the derivative of e to the ax. So, ax is like my g function, and e to something is my f. So I take the derivative of e to something. Well, that's just e to that something back again. And then I go inside and take the derivative of ax, and that's just a, so I multiply them out.
