 Later on, we'll want to form Taylor series when we don't know the function, and we can do this as long as we know something about the derivative. For example, suppose f' of x is f of x squared for all x, and we know f of 3 is equal to 5. Let's find the first four terms of the Taylor series for f of x centered at x equals 3, and assume that f is continuous and continuously differentiable. So, we know what our Taylor series will look like in general, and if we center it around 3, our Taylor series will look like. So, first of all, we need to find the value of f of x and all its derivatives at x equal to 3. Well, we already know that f of 3 is equal to 5, which gives us the first term in our Taylor series. Now, we need to find f' of 3. If only there was some way we knew what f' of x was. Oh wait, we know what f' of x is. We know that f' of x is f of x squared for all x. So, we have f' of x, well that's f of x squared, and at x equal to 3, this will be 25, and so that gives us the second term in our Taylor series. We need to find the second derivative at 3. Fortunately, we're given our second derivative, um, wait a minute, no, we don't have our second derivative. Now, could we find our second derivative if we don't know what our function is? Well, we do know something about the first derivative. Let's see if we can do that. Well, our first derivative is f of x squared, and remember the second derivative is just the derivative of the derivative, and so we can differentiate. So, this goes back to the chain rule. This is a something squared, so our derivative will be 2 something times the derivative of our something. Let's go back where you found them, and we can simplify, since we know what f' of x is, and so we know what our second derivative looks like, and at 3, our second derivative has value, which gives us the next term in our Taylor series. Well, now that we know that, we know how to find the third derivative. It'll be the derivative of the second derivative, and we found that our second derivative is 2 times f of x cubed. And so our third derivative will be evaluating this at x equal to 3, which will give us the fourth term in our Taylor series. And we can continue to find as many additional terms as we want to, and get a series expression for our function, even though we don't have an algebraic expression for it.