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$f(x)=\left(7 x^{3}\right)\left(3 x^{4}-9\right) .$ Find $f^{\prime}(x)$ in two different ways.

$21 x^{2}\left(7 x^{4}-9\right)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

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.

02:07

$f(x)=\left(x^{2}+2\right)…

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$f(x)=\left(5 x^{6}+2\righ…

01:54

$f(x)=x^{6}\left(2 x^{4}-3…

02:17

Find $f^{\prime \prime}(x)…

02:49

since this problem wants you to find the directive two different ways, Theo. Easiest way is to distribute seven X cubed into the problem. Three next to the fourth minus nine. So they define the function as this. And so the whole premise is you could rewrite. This is 21 7 times three x to the seventh because you add the exponents minus 63 um, sometimes nine x to the third. So if you use a calculator because I'm not very good at doing 21 times seven eso now you can find the drift of of this. We do this in green as prime of X. What equal? 147 and you subtract one from your ex on it. Same thing is 63 times three. I could do that in my head, though 189 x to the second. So this is one way of finding the directive. The other way is by using the product rule where you take the derivative of the left side, which would be 21 x squared. You leave the right side alone and then plus you leave the left side of line times the drift about the right side, which we 12 X to the fourth. Now, a teacher might say Okay, this is good. Nothing to stop right here. Or we could do is you could distribute this in and combine like terms and oh, I don't know what I did there. This the bomb derivative here, you know? Yes. Attract one from the Jackson area. Onda. What you'll see is you get the same thing 147 x to the six minus 189 X squared. Um, and there's so many other ways of writing as well that it's really up to you where you want to stop just to make sure you don't make a mistake like I do there. Um, so there you have it two different ways.

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