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Find: (a) $f g(x)$ ) and; (b) $g(f(x))$.$$f(x)=x^{2}+2 x+1, g(x)=\sqrt{x}.$$

(a) $x+2 \sqrt{x}+1$(b) $|x+1|$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

Idaho State University

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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So we're doing the composition function on this problem, and what they're getting into practice is to see that you know, the order does matter squared plus two X plus one. Um, now, I don't know how many people would notice this, but you could actually factor this problem as X plus one X plus one. So that is the same thing is X plus one squared on There is when I bring that to your attention is because if you look at G of X is equal to the square root of X, um, as we jump into part A where we do of G of x e don't know if this will be as important, because if you were toe do this problem, Um, you're supposed to do g of X first, which is the square root of X. So what I'm doing is I'm replacing G of X and the problem with square root of X, and then what I can do whether you go to the top equation this equation and replace the excess and that problem squared of X squared plus two square root of X um, plus one or if you go to the bottom equation. Replace this X with the square root of X plus one squared. You do get one of those as being the correct answer. Now, if you notice this top one, you can actually deduce, because we can only square positive numbers that this answer These will actually cancel each other out so you can just leave it as X plus two rude X plus one. Whereas if I go to Part B and we do G of f of X, we can actually plug in any number we want for the X function, which is that X Plus one squared. And then we're going to do the G of that. Now again, the domain is any all real, so we can play in anything we want for X. But we can Onley square root positive numbers. And that's why when we do this problem, we want when we simplify and we want to cancel out the square root in the square that we can only write this as the absolute value of X plus one as our answer on that one. So that's how I would write my two answers circled in green

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