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Given $f^{\prime}(x)=6 x^{5}\left(x^{6}+5\right)^{10},$ try to find $f(x)$ by guessing.

$$\frac{1}{11}\left(x^{6}+5\right)^{11}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Campbell University

Oregon State University

Harvey Mudd 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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eso We're giving them derivative in this problem and they are actually asking you to just guess what the original function is when you don't really need to guess the answer. Um, it makes perfect sense as soon as you do the problem X to the six plus five to the temp. And the reason why I really don't need to guess is, uh because I think you probably knows in this problem, um, that the derivative of this sorry, the derivative of this is this. So basically, what it's saying is, just think about if you left X to the six plus five alone, what you'd have to do is add one to the exponent to go backwards. And the whole reason for that is think about if you, uh, you know, move this 11 in front. Then you subtract one from that. Excellent to give you 10, which is why we want to add one to the exponents. Now. The other piece I'm mentioning is that the derivative of X to the six plus five is equal to this piece. So if you were to do the chain rule, you multiply by the derivative of the inside. Um Yeah. So that that is taking care of the problem, though. If you notice is I drew this arrow. Bring that 11 in front. Um, so if you did that, you would have 11th here. Well, we don't have an 11 up here in this problem, So let's think about going backwards instead of multiplying by 11. How do we undo that? That would be through dividing by 11. My God. So what makes the most sense is if you had 1/11 uh, extra the six plus five to the 11th hour. Um, And if you found the derivative of this, I think it's pretty clear that you would get this answer because the 11th would cancel and be now to the 10th Power times the derivative on the inside, which is, uh, matches up perfectly. So this is correct.

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