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Make a careful sketch of the graph of $ f $ and below it sketch the graph of $ f' $ in the same manner as in Exercises 4-11. Can you guess a formula for $ f'(x) $ from its graph?

$ f(x) = \ln x $

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

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.

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this problem. Number eighteen of the Stuart Calculus eighth edition, Section two point eight Make it careful. Sketch. With the graph of F and blow, it sketched the graph of Prime in the same manners and exercises for through eleven. Can you guess a formula for F prime of X from its craft and the function F is from X equals seven max so we can plot Ellen Rex as we're familiar with dysfunction. It has the shape, whereas expert's negative infinity or as experts zero from the left from the right, the function of purchase negative. Infinity as experts is infinity. The function approaches infinity, and another interesting point is that as ah at X equals one in the function, Ellen of X equals zero two. We can use this information to determine with the graph of Prime should look like and when we do that by using on the slope of the tangent lines to every at every point of dysfunction. So one of the easier ones scene is that let's say this is a slip of the tension line. At this point, as we get closer and closer to Texas is impunity. This function, uh, appears to level off the slopes decrease and approach zero. So we can show that here that eventually the slopes, our may decrease towards zero. And what we see here going towards ex zero from the right, see that the spokes become more and more steep. One more negative are more, more positive. And therefore, as X apprentice here from the rain, disfunction will be going towards positive infinity. And if we were to exactly figure out what's hope is here at X equals zero are at X equals one. We ended up determining that soap is indeed one here. X equals one. So at X equals one plus look is one. Thats the point one one and that's a character's a point here that function will need to go through. So this is the general shape of what crime should look like. And if we were to think kiss dysfunction is similar to the function one over X, meaning that we expect the derivative of Alan X to be similar to this Formula one a rex. And it is indeed true. The derivative of Alan X is one of wrecks on, so we were able to determine the shape of the derivative by using any slopes of the tension lines at each of the point of dysfunction, F equals Eleanor IX.

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