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Short segments of the tangent lines are given at various points along a curve. Use this information to sketch the curve.See Figure 11.

$$\frac{1}{2 \sqrt{x}}+2 / x^{2}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 1

Slope of a Curve

Derivatives

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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.

00:43

Short segments of the tang…

04:51

Find an equation for the t…

03:10

04:01

03:00

here. We're gonna be using the sum rule and power rule of derivatives in order to find the derivative of F of X is equal to the square of X minus two over X can see written in blue, the quick summation of the sum rule and the power rules you can follow along as we go. Let's start with our F of X, which is equal to square X minus two of her ex. Let's simplify this or not simplify, but let's rewrite it so that it's a little bit easier to work with. Rewriting it as F of X is equal to X to the one half because that is equivalent to the square root of X. Go ahead and subtract instead of two over X. Let's make this two X to the negative one. Because remember when you move a negative exponents to the denominator, it becomes positive. So it's almost as if this were a one here, which means if we do the room first and move it from the denominator to the numerator will get a negative. This just allows us to better use this power rule to find the derivatives of finding the derivative we have f Prime of X is equal to Let's go ahead and use the power rule here so you can see we have one half is our end. We can multiply that by what we have out in front. So it's just which is just a one that gives us one half X and then doing end minus one gives us a negative one that's going to be the derivative of our first term. Doing the second we're going to do the same thing with the power rule. Multiply negative two by negative one gives us positive two X and the negative one minus one gives us a negative too. Is that right? There would be our derivative, but we can simplify it a little bit further. Simplifying this we end up with F Prime of X is equal to one all over two x What I did here because we have the one half I moved to to the denominator and like we did before when we changed our route x two X to the one half. I just reversed this essentially at this point. But again, this was a negative one half, so that puts the square root of X into the denominator. Now our second piece, we're going to move that negative exponent down the denominator as well, giving us two on top and X squared on the bottom, and that would be our derivative of square root of X minus two over X.

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