Based on the graph of f ( x ) f\left(x\right) f ( x ) , describe the graph of the derivative f ′ ( x ) f^{\prime}\left(x\right) f ′ ( x ) at the point x = − 1 x=-1 x = − 1 .

So in this problem, we're told based on the graph f of x, describe the graph of the derivative f prime of x at the point x is equal to negative one. Okay. So in this situation, we have this graph that we're given to us, f of x, but we're asked to describe the derivative at one specific point. Now because of that, I'm not going to draw out the entire graph here. So I'm not going to draw, like, an entire curve. I'm just gonna focus on the point of interest that we have. So we'll have our derivative over here, which will have an x and y axis, and this is gonna be some function f prime of x. Now what I need to do is focus on how the graph and tangent lines behave at that point right there to see what our derivative is going to look like. So if we're focusing on this point of negative one, well, what would happen if I drew a line tangent to this point? Well, notice if I do this and I try to get the best line I can, I end up getting a flat line? A flat line has a slope of 0. So what exactly does this mean for our derivative graph? Is this gonna be some point that we have above the x axis, below the x axis? Well, where is it gonna be? Well, it turns out neither. It's actually going to be on the x axis. It's on the x axis because it has a slope of 0, so its output is gonna be 0 as a derivative, which means that we're just going to be right there on the x axis. So because we're on the x axis, we can see that that's where how our graph is going to behave for the derivative. So on the x axis would be the solution to this problem. And that is how you can solve these types of situations where you're looking at a graph at a specific point, and you need to figure how the derivative is going to behave. If we got a positive slope, it would be above the x axis. If we got a negative slope, it would be below. But if we get a flat line or a slope of 0, it's on the x axis. So that is the solution. Hope you found this video helpful.

2. Intro to Derivatives

Basic Graphing of the Derivative

Calculus

2. Intro to Derivatives

Basic Graphing of the Derivative

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Multiple Choice

Based on the graph of $f\left(x\right)$ , describe the graph of the derivative $f^{\prime}\left(x\right)$ at the point $x=-1$ .

Above the x-axis

Below the x-axis

On the x-axis

Verified step by step guidance

To determine the behavior of the derivative f'(x) at x = -1, we need to analyze the slope of the tangent line to the graph of f(x) at this point.

Observe the graph of f(x) at x = -1. Notice that the graph has a horizontal tangent line at this point, indicating that the slope is zero.

The derivative f'(x) represents the slope of the tangent line to the graph of f(x). Therefore, if the tangent line is horizontal, f'(x) = 0 at that point.

Since f'(x) = 0 at x = -1, the graph of the derivative f'(x) will intersect the x-axis at this point.

Thus, the graph of the derivative f'(x) is on the x-axis at x = -1, indicating that the slope of f(x) is zero at this point.

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Struggling with Calculus?

Based on the graph of f(x)f\left(x\right)f(x), describe the graph of the derivative f′(x)f^{\prime}\left(x\right)f′(x)at the point x=−1x=-1x=−1.

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Basic Graphing of the Derivative practice set

Based on the graph of $f\left(x\right)$ , describe the graph of the derivative $f^{\prime}\left(x\right)$ at the point $x=-1$ .