#Integral and differential calculus plus#

OK to take that derivative, I notice the t is appearing there in the usual place, and it's also inside the integral. And that's why I do it here because it uses all the rules. OK so I want to take the derivative of that. But also, it would be nice, I just think if you plug that in, to that differential equation it's solved. What as we created- as we derive the formula- well it won't be wrong because our derivation will be good. So when I see that equation and we'll see it again and we'll derive this formula, but now I want to just use the fundamental theorem of calculus to check the formula.

#Integral and differential calculus plus#

So with this, that solves the equation dy dt equals y plus q of t. That's a remarkable formula for the solution to a basic differential equation.

So it's a function of t then, time, It's the integral of this, e to the t minus s. The function I have in mind, I'll call it y, is the interval from 0 to t. And let me show you the function I have in mind. OK can I apply that to a really- I see this as a key example in differential equations. It says that if you take a function, you integrate it, you take the derivative, you get the function back again. I'll use the fundamental theorem right away. I won't discuss that fundamental theorem, but it certainly is fundamental and I'll use it. I can- I'll change that dummy variable to t. Here is y and the integral goes from 0 to x I don't care what that dummy variable is. The derivative of the integral of a function is this. And it says that one is the inverse operation to the other. So the fundamental theorem involves the derivative and the integral. That's really- that it's chains of functions that really blow open the functions or we can deal with. The derivative of this- of that chain of functions, that composite function is the derivative of f with respect to g times the derivative of g with respect to x. Who can remember that?Īnd above all, the chain rule.

The derivative of f plus g is the derivative of f plus the derivative of g. Secondly, out of those few specific facts, you can create the derivatives of an enormous array of functions using the key rules. And then the inverse function related to the exponential is the logarithm. That's the wonderful equation that is solved by e to the x. The derivative of e to the x is e to the x. Above all, the derivative of e to the x, which is e to the x. The derivative of x to the n, the derivative of sine and cosine. And the first thing is, you really do need to know basic derivatives. So I'm not saying forget all those, but just focus on what matters. And I'm going to show you what I see, and it's not everything by any means, it's some basic ideas, but not all the details you learned. Differential equations is the big application of calculus, so it's kind of interesting to see what part of calculus, what information and what ideas from calculus, actually get used in differential equations. And that I think it's worth thinking about what we know.