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Use the appropriate rules to determine the derivative.$$y=3 x^{2 / 3}-3 x^{4}+2 / x^{2}-7, \text { find } y^{\prime}$$

$$\frac{2}{x^{1 / 3}}-12 x^{3}-\frac{4}{x^{3}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

Harvey Mudd College

Baylor University

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.

02:42

Use the appropriate rules …

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Use the rules of different…

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Use the General Power Rule…

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Use the Chain Rule to find…

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Find the derivative of the…

Alright, This problem is pretty straightforward because, um, all you have to do is is find why prime And if we're given, why equals? All we have to do is find the derivative. Now there's a little bit of algebra manipulation to make this problem easier. Um, but most students should be able to do this in their heads. If you cannot do this in your head, it's not that big of a deal. I would just advise you to get it figured out. Um, minus two over x squared. Look. Oh, wait, that's a plus. And then minus seven and may, it might be easier to do a little bit of of understanding. First, I'm not going to rewrite the whole thing, But this little piece right here is the same thing is saying to x to the Negev second power. So then, when I do the derivative and you multiply the expo about what's in front, uh, I'll go ahead and just rewrite it. Because some students are struggling with subtracting one from the exponents, it might make more sense to write three thirds. So you're denominators already the same. Now, the next one, I would anticipate you do it in your head four times three is 12. Subtract one from your exponents. Now, doing the next one. It makes perfect sense. Then it becomes negative. Four x to the neck of third power in the drift of of a constant is zero. So no need to write that. So this preliminary work can help you identify that. Okay, these threes canceled. And now two thirds minus three thirds is negative. One third. You know, this one's good. Um e mean, this other one is good as well. Just some students forget that Subtracting one from a negative. Um, yeah. Um, you know, we're getting more negative. So native to minus one is negative Three. Um and you know, depending on who your teacher is, some teachers let you leave your answer like this. Other teachers might say, Well, go ahead and rewrite us the cube root of X in the denominator because of the negative exponents, you know, minus four over X cubed. Something like that.

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