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Find $f^{\prime}(x)$ if $f(x)=\sqrt{2 x^{3}+3 x+2}$.

$$\frac{3\left(2 x^{2}+1\right)}{2 \sqrt{2 x^{3}+3 x+2}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Missouri State University

Campbell University

Baylor University

University of Michigan - Ann Arbor

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.

04:49

Find $f^{\prime}(x)$.$…

03:08

02:59

Find $f^{\prime \prime}(x)…

04:03

Find $f^{\prime}(x)$ if $f…

eso we need Thio, Take the derivative of this giant square root function. Uh, two X cubed plus three X plus two. Yeah, and we are going to use a change rule. I'd like to have my students rewrite the square root to the one half power. Uh, you don't have to and and honestly, the better you are not doing this, you will get better. Um, you know, probably understand the math a little bit more. But anyway, what we have is this inter function. So I like to have my students point out the inner and outer function. So when we take the derivative using the chain rule, what we do as we do the outer functions derivative first, what I'm talking about bringing that one half in front. And then it's to the negative one half power because you have to subtract one from you're exploits and what you do with the change roles, you leave the inner peace alone. But then when you multiply by the derivative of the inner peace So the derivative two x cube is six x squared, the derivative of three x three derivative of 20 So we don't write that down aan den from here. Some teachers actually let you leave your answer alone. Just leave your answer like this. Um otherwise teachers might say, Oh, you can simplify that. And it's not that bad. It really isn't where you write it as a giant fraction because the two goes in the denominator. The negative exponents sticks it into the denominator. And to the one half power, just like I did at the beginning is the square root. And you could factor this, but I would just leave it alone because nothing canceled. So there's really no benefit of factoring. So this is another option for you for a correct answer. So both circled in green are correct.

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