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Find the derivative of the function.$ F(t) = \frac {t^2}{\sqrt {t^3 + 1}} $

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$\frac{2 t\left(t^{3}+1\right)^{1 / 2}-\frac{3}{2} t^{4}\left(t^{3}+1\right)^{-1 / 2}}{t^{3}+1}$

00:30

Frank Lin

04:23

Heather Zimmers

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 4

The Chain Rule

Derivatives

Differentiation

Campbell University

University of Michigan - Ann Arbor

Idaho State University

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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Alright. Our goal is to take the derivative of this function. This function is a quotient. So we'll end up using quotient rule. But before we do it, I'm gonna go ahead and rewrite the bottom as t cubed plus one to the one half power. And now we're going to do product rules. So just remind you how product roll works. If I want the derivative of something with say you on top over V, then we do the derivative. Let me actually make this a little bit easier to read. I'm just going to do just U. And V. But both of these are a function of X. So I'll do the derivative of the top times of bottom minus the derivative of the bottom, times the top all over the bottom squared. So that's what we're gonna do. And this is known as quotient rule. Um Okay, so let's go ahead and get to it then. Alright, so f prima, t. Then our derivative will be derivative at the top. So I get to T times the entire bottom. Now I need to subtract and take the derivative of the bottom, which by power rules 1/2 T. Q plus one to the minus a half. And then I still have to to change roles. So I multiply by the derivative of the inside which is three T squared. And I still have to multiply by the top. So there's a lot there. Okay, all over the bottom squared. So T Q plus one. If I square square root, I just get the privacy without any power. So um that's looking good. So let's clean this up. Just a tad um Just see what we have kind of here looks like we have two T Square Root T. Cubed Plus one. If I clean all this up I get -3/2. T to the fourth. All over T cubed Plus one in the Square Root. All over T. cubed plus one. Alright we can keep going and go further but I think this is sufficient for you to know how to set up the derivative quotient rule and then whatever cleanup you'd like to do after this, you're certainly welcome to do. But there we have it we have solved for our derivatives. So hopefully that helped to have an amazing day.

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