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Find $\frac{d}{d x}(\sqrt{2 x+\sqrt{3 x+\sqrt{4 x}}})$.

$$\frac{2+\frac{3+\frac{1}{\sqrt{x}}}{2 \sqrt{3 x+2 \sqrt{x}}}}{2 \sqrt{2 x+\sqrt{3 x+2 \sqrt{x}}}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Oregon State University

Baylor University

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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all right, We're just gonna jump right into the derivative. And this is using the chain rule over and over again. Uh, so please stick with me, because this is gonna take a while. When we had the square root of two x plus square root of three x plus the square root of four X, uh, which is basically just saying, Hey, we're going to do the change, will inside of the changes inside the changing room. And I like to rewrite this problem as a four X. That's to the one half power. And you have this three x plus That's to the one half hour, because that's what the square root is. And then you have this two X, and that is to the one half hour with a close parentheses there. Hopefully, my parentheses makes sense. So anyway, way have to start with the outermost square root. Uh, which is this one half needs to be brought in front. They're looking at one half. Everything inside stays the same, and it's now to the negative one half hour eso I'm going to revert back to the original problem how it was written as two x plus our Route three x plus, Route four X. But then you have to multiply by the derivative of the inside, which now we're done with that exponents, the derivative of the inside is two plus Well, now we needed to the derivative of and maybe I should have underlying here of this piece, which is bringing this one half in front. So let's ah, let's do that one half. She's gonna put some parentheses in here because now that's the negative one half power. I'm gonna revert back to the three x plus square root for X. But now we need the derivative of that inside which, if you look at that, it's three. I guess I should underline in a different color. Look at this piece now. Eso It's three plus. Now I'm bringing the one half in front of that on half the four X is now to the negative one half power and then times the derivative of the inside of just times four. Now, you can clean this up as much as you want, but no matter what you do, you're gonna have an ugly answer. Um, one suggestion is maybe just say, like, if your teacher makes you do something. Let's simplify half of four after forced to. At least you did something. But there's really no rhyme or reason why you should have to simplify this anymore. This should be good enough.

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