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Determine the derivative at the given point on the curve using equation (2).$f(x)$ as defined in Exercise 5.

-5

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Campbell University

Harvey Mudd College

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.

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Determine the derivative a…

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Express the derivative of …

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Find the derivative of the…

01:41

here, we're gonna be using the formula that you see boxed in in order to determine the derivative of F of X equal to negative two X squared plus three X plus three. Once we find that derivative, we want to figure out what it's equal to when X is equal to two. So let's start by finding each piece of our equation. Let's start with F of X plus H. Working with the numerator F of X plus H is just going to be equal to whatever x by inserting expose age into our f of X. So that's going to give us negative, too. X Plus H squared was three x plus age plus three, so you can see that each of these is just where we had an X and R F FX function. Let's go ahead and simplify this, giving us negative two times. Let's pull apart these Expos H squared so that we can then combine them even further, plus distributor three x into our expose H plus three, giving us a negative two. Let's foil all of this out, giving us X squared plus two X h plus H squared plus three eggs plus three age plus three. That's my bed. It looks like I wrote in eight right there. That should have been an Urban X. Should have been an age. Go ahead and distribute our negative two. Now given us negative two X squared. Plus, actually that will be a minus four X h minus h squared. We still don't have plus three x plus three h plus three. Taking a look. See if we can combine any like terms doesn't look like we can quite yet. Now, keeping in mind we still to subtract ffx from all of this f of X, which is negative. Two. X squared plus three x plus three. Let's go ahead and distribute the negative to each of these terms to give us a positive two X squared minus three X minus three And that's still with everything we just had here. We'll just bring it down to X squared is four x h minus H squared plus three x plus three age plus three Simplifying here. There are definitely some lake terms we can combine to cancel. Looks like our negative two X squared cancels with our positive two. X squared Our positive three x and negative three x Cancel Positive three and negative three. Leaving us with negative four x h minus H squared plus three age Perfect. Keeping in mind now that we still have to divide all of this by H, we don't want to forget about that and our function here. We want to simplify even further, and it looks like we can factor out an H and R numerator pulling on an H. We're left with negative four x minus h plus three all over age, which allows these two to cancel getting rid of our denominator, pulling us out of fraction format. And we are left with a simplified version where F prime of X is equal to the limit. Has h approaches zero of negative four x minus H plus three here. If we plug zero in four h, we end up getting negative four x Plus three as our derivative. That's what prime of X is equal to, but ultimately we want to know what it's equal to. An X is equal to two, so let's plug in for F prime of X we want to do to, which would be negative. Four times two plus three simplifying. We get f prime of negative two is equal to it. Looks like we'll get a negative. Eight plus three gives us five negative five. So the derivative of negative two X squared plus three x plus three when X is equal to two is equal to negative five.

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