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Find two functions $f$ and $g,$ whose composition $f(x)$ ) will result in the given function $h(x)$.$$h(x)=\left(5 x^{3}-7 x+3\right)^{3 / 5}.$$

$$f(x)=x^{3 / 5}, g(x)=5 x^{3}-7 x+3$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

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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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Okay, so we're doing the decomposition of this function. And what we need to get is other form f of G of X. I always forget which some books to gff FX and that zone wants this. Okay, so what we have is five x cubed, minus seven X plus three, and we're taking this to the three fist power. So the way I like to decompose thes functions is by identifying parentheses or some sort of inter function eso the parentheses air good indicator for the inter function with the outer function, the outer function being taken to the three fist power eso Then from here, what I can do is say Okay, well, the inter function needs to go on the inside of my decomposition. So that tells me that g of X, the parentheses would be the five x cubed minus seven x plus three. Um, now, the reason why my students get confused is Then when I look at ffx, which is the outer function, they want to do something special with the inside. Well, this inside is no longer there. So sometimes what I'll do is I'll just put a giant X in here like it's gone, and it helps Some students recognize that. Okay, well, that X is holding the spot of the function. That's to the three fist power. Now some people might actually rewrite. This is the fifth root of X Cubed. That's just getting fancy, though. But it is true. And if you were to do the composition function of F g of X, you get the correct answer. Um, you know, by putting that together s I'll circle it and move on. This is the right answer. Hopefully, my explanation is making sense.

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