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Use the appropriate rules to determine the derivative.$$w=32 v^{1 / 4}-\frac{16}{v^{2}}+7 v^{2}+2, \text { find } \frac{d w}{d v}$$

$$\frac{8}{v^{3 / 4}}+\frac{32}{v^{3}}+14 v$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

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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Yeah, So in this problem, the phrase D W. D. V is basically just saying, Hey, you're taking the derivative and it makes the most sense to rewrite the problem with rational exponents. B to the 1/4 now to the 1/4 is already rational. But what I'm talking about is the 16 over B squared and makes more sense to write B to the name of second power. Um, in the seven b squared is fine, and the plus two is fine as well. Eso this the main thing that you would want to rewrite because then, as you do the derivative, which is what we're asked to do, and you bring that 1/4 in front well, that's the same thing is saying 32 divided by four, which should be eight You and then you have to subtract one from that explanation. Then, if you're struggling with subtracting one, just get the same denominators of four force, so one minus four is negative. Three. Force. This is going to become 32 because as you multiply a negative times, the name is positive. And then when you subtract one from negative to get negative three seven times to give me 14. Subtracting one from the explosion gives me one on, then the derivative of the constant zero. So no need to write that down now. I would let my students leave their answer like that. It looks like the answer key would prefer to write all the negative exponents back into the denominator, which is true. Um, so there's nothing wrong with this answer either. From there you have it because

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