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Find: (a) $f g(x)$ ) and; (b) $g(f(x))$.$$f(x)=x+5, \quad g(x)=x^{2}.$$

(a) $x^{2}+5$(b) $(x+5)^{2}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Missouri State University

Oregon State University

Harvey Mudd College

Boston College

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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01:39

All right. So we're just practicing the composition function on this problem where the they define ffx as X plus five and they defined G of X is X squared. And for part A, they want you to practice finding f of g o X. Now the way I think of this is ah, what's inside is the function that gets done first. So that means we're taking G of X and replacing it into this problem. So we're looking at F of replacing G of X with X squared and then from here. If I'm tasked with finding what f of X squared is, that means to take the F function. I just think about taking the X value out and replacing X with X squared. And that's the answer. They're looking for it for part a eso in part B, where they switch things around. It is a very, very different thing. Um, so again, if you were to do, I'll do this in red because we're doing the inter function first. That's ffx is X plus five. And what you're doing then is going to the G function, which is Ah, I'll switch Colors again is being squared, but notice. What I'm doing is I'm replacing the X in the problem with X plus five. So I have to replace the X in this problem with X plus five. Now, this is an accepted points for R B. Otherwise, some teachers will. May I ask you to actually foil that out? There's really not a benefit to foiling this, but I do want to point that out that if you distribute everything you have X squared five X plus five X is 10 x on, then five times 5 25. But like I said, this is a better answer, especially as we move into the derivative here shortly.

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