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

See Sketch.$f^{\prime}(x)=e^{x}$

Calculus 1 / AB

Chapter 3

Derivatives

Section 2

The Derivative as a Function

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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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Make a careful sketch of t…

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$16-18$ Make a careful ske…

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Simplify $f(x),$ and sketc…

Okay, so here we want to start by a graphing Ethel. That's equal to e to the power. That's okay. So if we have our ground, if we were to take eat of the power of zero, that's gonna be equal to one. So we know that this will be equal one at zero for negative numbers, the more negative it gets, right? What we end up with is, you know, if it's negative seven, it would be equal to one over. You are seven rights. We're gonna get bigger and bigger values at the bottom, which means it's a smaller and smaller fraction that never quite gets to zero s. We know this is gonna just keep going down and getting closer and closer zero but never quite getting to zero. And then similarly, for Exit E being greater than zero, we're just gonna get this nice exponential number that's gonna keep growing. So this graph looks probably wanted blue there. We'll just keep going up like this. That's our graph, you the X. And if we draw the derivative the way we have, where we kind of okay, just borrow from from one. The graft that we have we can see that is positive the whole time. So it's going to stay above the X axis, right? And if we take our tangent lines about here, we have roughly a 1 to 1 tanja line. Um, here it gets even higher, right? So and then here it's getting closer and closer to zero. So at this end, it's close to zero. Here it's roughly equal toe. Wanna wonder one line and then here it's greater than one. And so if we plot that, what looks the exact same right? So So we can guess, then that the derivative of either the X is e to the X.

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