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Find two functions $f$ and $g,$ whose composition $f(x)$ ) will result in the given function $h(x)$.$$h(x)=(2 x-1)^{3}+5(2 x-1)^{2}+5$$

$$f(x)=x^{3}+5 x+5, g(x)=2 x-1$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

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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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we are doing another decomposition function and what I mean by decomposition, we're gonna decompose h of X to be of G of X. They're gonna find two functions that would make that equation. So as I copied down the problem, what you want to do is look for parentheses. Most cute. Sorry. Five. And once you can identify a set of parentheses that are repeated in the problem, uh, then that is probably what your inner function will be. So what I tell, my sense is look for parentheses. So we have parentheses here for G of X. So that's my inner function. So what I'm doing is I'm looking for parentheses inside of H Becks, which can represent my inter function. So what I'm deducing from this is that g of X is equal to two X minus one. So then what I can do is is imagine that I take out that inter function. Okay, so I'm removing these pieces and, you know, we can put a giant X in there. I know my students get confused because we already talked about X. So how can you use another ex? Um, well, we're defining everything in terms of X. That's the best way I can answer this. So the outer function would be as if we remove the inner peace and replace the inner peace with X. So looking at X cubed plus five, replacing the inter function with X is now squared plus five. And these are the two answers that we want or the problem. And you can check that you're right because this is math. You always check your answer math by doing the composition function f of G of X and see if you get h of X and you will This is correct.

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