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Determine the derivative at the given point on the curve using equation (2).$f(x)$ as defined in Exercise 7.

12

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

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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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02:57

all right. Using the function for finding a derivative here, boxed in, we're going to be finding the derivative of execute. Once we find that derivative, we're gonna plug in X equals two to determine what that derivative is when X is equal to two. Let's start by finding what our numerator of this function will be with f of X plus H 39 f of x plus h is equal to X plus h cute. And we're gonna might subtract our FX, which is just execute. Let's go ahead and pull apart our X plus ages so that we can further simplify explosive times. Explosive times explosive charge Still remembering ar minus X cubed. Go ahead and foil these out right here, giving us X squared plus two x h plus h squared That is all still going to be multiplied by another X plus h. So our minus execute Go ahead and multiply our X by each one of these terms giving us execute plus two X squared H plus X h squared Now multiplying our h out through these, we're gonna get plus X squared age plus two x h squared plus h cubed and we still have our minus X cubed down here. Now we can start to combine like terms and hopefully cancel some of these terms out to shrink this a little bit because we've got a lot going on right now. See our ex cubes cancel out and it doesn't look like anything else. Well, but we should be able to combine some here. So we've got looks like we have an X squared each year and another x squared each year. Combining those we get three x squared age and we have another X h squared and H squared. They're giving us three x h squared plus the H cube that we're left with. That looks like it does. Simplified as that's going to get. Unless what we could do if we wanted to, we could pull out an H on the top. This will help us out in our next step. Plot an age to give us three X squared plus three x h plus h squared. Now we still have to divide by this h you see up here and our function we're following dividing all that by H. Now we're able to cancel out these ages, so you can see the value and pulling out the age on the top that we were able to simplify it down to get our final answer. Where F prime of X is equal to the limit as a church approaches zero of three x squared plus three x h plus each squared now to find what the derivative is. All we need to do is find salt for this limit, which means plugging zero. And for each of these beaches, if we do that, zero squared is just zero. So that cancels three X times zero is also zero. So that cancels giving us our derivative of F of X is equal to three x squared Now. If we wanted to solve for what the derivative is, when X is equal to two, we would just plug that in three times two squared. That gives us an answer equal to 12. So when F Prime is equal to two, it's equal to 12

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