We could try to, we could try to simplify this a little bit or rewrite it in different ways, but there you have it. So this part right over here is going to be cosine of x. This is this right over here, and then what's g prime of x? G prime of x, well g prime of x is just, of course, the derivative of sine Two sine of x, and then minus one, minus one. Here is going to be equal to everywhere we see an x here, we'll replace with a g of x, so it's going to be two, two times sine of x. What h prime of x is, so I'll need to do this in another color. And so what would that be? Well, we already know Because if this is true, then that means that capital F prime of x is going to be equal to h prime of g of x, h prime of g of x times g prime of x. Now why am I doing all of that? Well, this might start making you think about the chain rule. So you replace x with g of x for where, in this expression, you get h of g of x and that is capital F of x. Is if we were to define g of x as being equal to sine of x, equal to sine of x, our capital F of x can beĮxpressed as capital F of x is the same thing as h of, h of, instead of an x, everywhere we see an x, we're replacing it with a sine of x, so it's h of g of x, g of x. Instead of having an x up here, our upper bound is a sine of x. But this one isn't quiteĪs straightforward. Theorem of calculus that h prime of x would be simply this inner function with the t replaced by the x. Let me call it h of x, if I have h of x that wasĭefined as the definite integral from one to x of two t minus one dt, we know from the fundamental Just to review that, if I had a function, If it was just an x, I could have used theįundamental theorem of calculus. So some of you might haveīeen a little bit challenged by this notion of hey, instead of an x on this upper bound, I now have a sine of x. What is F prime of x going to be equal to? So pause this video and see Solution: In this example, we use the Product Rule before. Solution: Example: Differentiate y (2x + 1) 5 (x 3 x +1) 4. Example: Find the derivatives of each of the following. We differentiate the outer function and then we multiply with the derivative of the inner function. Upper bound right over there, of two t minus one, and of course, dt, and what we are curious about is trying to figure out Note: In the Chain Rule, we work from the outside to the inside. That we have the function capital F of x, which we're going to defineĪs the definite integral from one to sine of x, so that's an interesting
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