![]() And you really can't, this notation makes it These differentials, this d whatever, thisĭx, this d sine of x, as a number. And this is where it might start making a little bit of intuition. ![]() And then we're multiplying that times the derivative of sine of x, the derivative of sine of x with respect to, with respect to x. Taking the derivative of with respect to sine of x, with respect to sine of x. We're taking the derivative of, we're taking the derivative ![]() The derivative of sine of x with respect to x. Of x is two sine of x, and then we multiply that times So derivative of sine of x squared with respect to sine Of the outer function with respect to the inner. And so there we've applied the chain rule. Seen multiple times, is cosine of x, so times cosine of x. The derivative of sine of x with respect to x, we've Sine of x with respect to x, well, that's more straightforward, a little bit more intuitive. We say two sine of x times, times the derivative, do this is green, times the derivative of And it would've been just two x, but instead it's a sine of x. So we could view it as theĭerivative of the outer function with respect to the inner, two sine of x. So that's going to be two sine of x, two sine of x. Outer function, x squared, the derivative of x squared, the derivative of this outer function with respect to sine of x. Going to be the derivative of our whole function with respect, or the derivative of this Now, so the chain rule tells us that this derivative is Two times the thing that I had, so whatever I'm taking theĭerivative with respect to. Up here, the a's over here, I just replace it with a sine of x. ![]() What if I were to take theĭerivative with respect to sine of x, with respect to sine of x of, of sine of x, sine of x squared? Well, wherever I had the x's Now I will do something that might be a little bit more bizarre. 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
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