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$f(x)=x^{6}\left(2 x^{4}-3 x+4\right) .$ Find $f^{\prime}(x)$ in two different ways.

$x^{5}\left(20 x^{4}-21 x+24\right)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Baylor 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.

01:55

$f(x)=\left(5 x^{6}+2\righ…

02:07

$f(x)=\left(x^{2}+2\right)…

01:39

Find $f^{\prime}(x)$.$…

02:11

Find $f^{\prime}(x)$ at $x…

Okay, so we need to do this derivative in two different ways. The first way I'm going to show you, it's it's the same thing. You could have done a couple of units a couple sections ago where you would want to distribute, So this is what f of X is equal to. So as you distribute X to the six and to all those terms, you'll just get you just have to add your exponents. That's a rule with multiple with the same base. And then from there you could look at this and say, Oh, I confined that derivative pretty easily because you bring the exponents front multiplied by the coefficient, so that makes it 20 and subtract one from your exponents minus 21 because that's three times seven x of the six. Four times six is 24 thanks to the fifth. So this is a good enough answer now if you notice you know, if you're checking with the back of the book or whatever, um, you could greatest common factor that term out. Otherwise, what you can do is look back at this problem and do the product rule from the get go well What's the product rule? Yeah, find the product first. Here's the product left side and right side. So you take the drift above the left side. You leave the right side of home and then plus, you leave the left side alone times the derivative of the right side, which would be eight. Execute minus three now if you wanted to. So this is another answer. If you wanted to, you could distribute this in and distribute this in and you'll see it. You'll get the same. Answer is over here and again. If you want, you can create. It's coming factory term out that you don't have to So there you

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