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$f(x)=\left(5 x^{6}+2\right)\left(2 x^{4}+9\right) .$ Find $f^{\prime}(x)$ in two different ways.

$2 x^{3}\left(50 x^{6}+135 x^{2}+8\right)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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.

01:54

$f(x)=x^{6}\left(2 x^{4}-3…

02:07

$f(x)=\left(x^{2}+2\right)…

01:01

f(x) = (x - 2)^5 + 6. Find…

01:12

Find $F(2)$ and $F(5): F(x…

00:50

Find $f^{\prime}(x)$.$…

01:14

02:13

$f(x)=\left(7 x^{3}\right)…

01:45

help please

eso I'm gonna show you two different ways of finding this derivative. Um, kind of let you work from there and to actually the fourth plus nine cool. And they define this is ffx. So the first way I'm going to show you is if you were to foil this out. You know, distributing, I guess, would be a better way of saying that where you get 10 X to the 10th power, uh, plus 45 next of six and then distribute this to in here you get plus four X to the fourth plus 18 because now what you can do is take the derivative of that thing by following your old rules. You know, 10 times 10 is 100 extra the ninth. I'm using a calculator. I don't know. 45 times six is in my head, 270 x to the fifth on a four times 4. 16 x to the third and the derivative of a constant zero. So this is one way of doing the problem. Another way is by doing the product rule where the derivative of F is equal to you. Take the derivative of the left side which is 30 x of the fifth and you leave the right side alone and then plus, you leave the left side of room. I thanks to the six plus two times the derivative of the right side, which is eight x cute. Now, this is an acceptable answer as well. Depending on who your professor is, they might say, Oh, you can simplify that. But if you simplify this and distributed and then combine like terms he was going to get the same answer is right there. So I don't see a benefit to doing that. Um, it's up to you. You have two different ways.

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