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Find $d v / d t$ if $v=t^{2} \cdot \sqrt[3]{(2 t+8)^{2}}$.

$$\frac{8 \sqrt[3]{4} t(t+3)}{3(t+4)^{1 / 3}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Harvey Mudd College

Baylor University

Idaho State University

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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we have another product rule mixed with the chain role T squared, cubed root of 22 plus eight squared. Now, if I were redoing this problem, I would still would still leave the product in there. Can't really change that. But I would rewrite to t plus eight to the two thirds power Um, eso Then when we do the product rule and chain, bro for that matter. The derivative of the left side is just to t you leave the right side alone to t plus eight is to the two thirds power. And then plus, now you leave the t squared alone times the derivative of the right side. This is the chain rule where you bring the two thirds in front, uh, to t plus one stays the same. You subtract one from your experiments, so it becomes negative. One third power times the derivative of the inside to be two teams. Now, as far as cleaning this up, I wouldn't I would just leave the answer like this. This is like, when do you stop simplifying? I don't I don't have a good answer for you. Um, otherwise you can combine like terms. You know, But at what point do you stop? You know, cube, Root of two T plus a squared. Uh, the two times two is four t cute over three cube root of two T plus eight. You know, this is a good answer. Or you could just look in the book to see what they come up with. You might take it a step further and get the same denominator. Um, but it's really up to you. When do you want to stop? And I would have stopped a lot earlier then this work down here.

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