This is still going to be equal to two a. Now, what if I were to take theĭerivative with respect to a of a squared? Well, it's the exact same thing. To x, what do I get? Well, this gives me two x. The derivative operator to x squared with respect If I were to ask you what is the derivative with respect to x, if I were to just apply Now, what I want to do is a little bit of a thought experiment, a little bit of a thought experiment. Seem obvious right now, but it will hopefully, maybe by the end of this Into play every time, any time your function canīe used as a composition of more than one function. I'm going to use the chain rule, and the chain rule comes Is what is h prime of x? So I want to know h prime of x, which another way of writing it is the derivative of h with respect to x. Now, I could've written that, I could've written it like this, sine squared of x, but it'llīe a little bit clearer using that type of notation. It's equal to sine of x, let's say it's equal to sine of x squared. H of x, and it is equal to, just for example, let's say But as you see more and more examples, it'll start to make sense,Īnd hopefully it'd even start to seem a little bit simpleĪnd intuitive over time. And when you're first exposed to it, it can seem a little dauntingĪnd a little bit convoluted. Time you take the derivative, anything even reasonably complex. What we're going to go over in this video is one of theĬore principles in calculus, and you're going to use it any It can be shown that ƒ is holomorphic, and that ƒ'(z) = 1 - 3𝑖 for every complex number z. Furthermore, the product rule, the quotient rule, and the chain rule all hold for such complex functions.Īs an example, consider the function ƒ: C → C defined by ƒ(z) = (1 - 3𝑖)z - 2. For instance, the differentiation operator is linear. The process of differentiation of complex-valued functions defined on subsets of the complex plane shares many properties with differentiation of real-valued functions defined on subsets of the real numbers. This property may also be cast in terms of convergent sequences in U. If such a number L exists, we usually denote it by ƒ'(w). More specifically, to say that ƒ: U → C is differentiable at an interior point w in U means the following: there exists some complex number L such that for every real number ε > 0 there exists a real number δ > 0 with the property that for all complex numbers z in U with 0 < |z - w| < δ, we have |/ - L| < ε. Holomorphic functions are central in the theory of complex functions. If ƒ is differentiable on an open set U, one also says that ƒ is holomorphic on U, or sometimes that ƒ is analytic on U. If U is open, and if ƒ is differentiable at every point of U, we say that ƒ is differentiable on U. If the limitĮxists, we say that ƒ is (complex) differentiable at w, and we denote the value of this limit by ƒ'(w). Suppose further that ƒ: U → C is a complex-valued function defined on U, and suppose w is an interior point of U. Let C denote the set of complex numbers, and suppose U is some subset of C. Did you mean to ask about the differentiation of complex-valued functions defined on subsets of the complex plane? Such functions may (sometimes) be differentiated. Since a complex number in itself is a constant, its derivative is zero. They seem to get the point across very efficiently. You may want to review some of Sal's videos on derivatives - especially the ones where he graphs the derivatives intuitively. In sum, basically, the chain rule takes into consideration of how the functions within a function determine the function's slope at some input. That's the only time you will make sense of it all.

Even when the slope of f(x) is 0 at x = -2, we see the graph of the derivative crosses the x-axis at x = -2.Ī lot of this has to do with looking at a graphs of a function and their derivatives on the same graphing sheet. we see that the x-value produces an incredibly negative y-value, which is the slope of the function f(x) at the x-value of interest (sometimes called a). and we know the derivative is decreasing. Towards the top of of the parabola on the left side, it almost looks like a straight line.

then, if you graph 8x+4 on the same sheet of graphing paper. If you graph (2x+1)^2 you will see that it is a parabola. When you apply the chain rule, you're taking into account how the slope of the function is behaving by the influence of the internal variables.

Ap calculus chain rule quiz