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Given $f^{\prime}(x)=18 x\left(3 x^{2}+9\right)^{2},$ try to find $f(x)$ by guessing.

$$\left(3 x^{3}+9\right)^{3}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Missouri State University

Harvey Mudd College

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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01:14

I kind of like this guessing concept. Um, you know, But you can actually algebraic. We figure this out. It's not that difficult. But if you've learned anything from this unit, it's that Okay, what will probably happen is the function for the original function. F will be the inter function, which is that three x squared plus nine. Um, but then you have tow add one to your extra, because as you undo the problem, um, you subtract from your excellent so it makes sense. Toe add Well, you can double check that. This is the right answer by taking the derivative s. Oh, this is my guess. That's the right answer, and we'll fix it if it's not where you could bring that three in front, you leave three X squared plus nine alone. It's now to the second power because you subtract one from the accident. So so far so good. Well, the 18 X is not good yet, but when you take the derivative of the inside, it's time six x and now you can verify. Oh, yeah, well, three times six. That's 18. And you have that X. So we're good. If anyone's curious, the way to actually figure this out is through a process of, um, you know, kind of doing the same thing and noticing that the derivative of the inside is six x. Well, you can fix that six X by multiplying by some constant. And that's what I meant earlier when I said, Well, if it's not this answer, we can fix it by by figuring out what this constant is. But we don't have to, because this is the right answer, okay?

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