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We've talked about how to take a derivative and what it looks like compared to it's parent function, but we haven't discussed the graphical version. From the graph of a parent function, we can learn a lot about the derivative graph and vice-versa.
What does a derivative tell us?
A derivative tells us the slope of the tangent line or the instantaneous rate of change
- Check out the 'How do I take a Derivative?' article for a bit of refresher if you're confused on this
Here are the basic fundamentals
- d/dx = positive → parent function increasing
- d/dx = negative → parent function decreasing
- d/dx = zero → possible local min/max
Decreasing and negative
- a decreasing interval on the parent graph corresponds with a negative interval on the derivative graph
- This is because decreasing means the graph is 'going down' or in other words, the slope of the tangent lines are negative
Increasing and positive
- an increasing interval on the parent graph corresponds with a positive interval on the derivative graph
- This is because of the same reasoning as before. Since the parent graph is 'going up', the slope of the tangent lines are positive.
Similarly, we can reverse these roles and say that a negative section of the derivative means that the parent function is decreasing and a positive section of the derivative means that the parent function is increasing.
(Local) Maximums and Minimums
On a graph, it's extremely easy to see where there are maximums and minimums on the parent graph. But, how do we find one on the graph of the derivative?
- In this graph, the green graph represents the parent function while the purple graph represents the derivative.
- Here, we can clearly see that the parent graph has a maximum and minimum around -1.3 and 1.3. But how do we see this without the graph of the parent function and only the one of the derivative?
To do this, a lot of your knowledge of derivatives and what they mean will come in to play. The graph of a derivative is basically just showing the instantaneous rates of change at each and every point of the parent function. Now, we have to translate that mumbo-jumbo into something we can understand.
Think of it like this. The derivative graph is basically just graphing the 'slopes' of tangent lines to the parent function. So, at a maximum or minimum, the 'slope' of the tangent line would be 0 because the tangent lines would be horizontal.
- For a maximum, we can see that as it gets closer to its peak, the slope of the tangent lines are positive, and as it comes down, its slope is negative.
- For a minimum, the same concept applies. As we get closer to the minimum value, the slope is negative and when we come back up, the slope is positive.
- Because of this, we can say that when looking at the graph of the derivative, there will be a maximum or minimum wherever the the graph crosses the x-axis.
- This means that on the graph of a derivative, a maximum means that the graph goes from positive, to zero, to negative. A minimum means that the graph of the derivative goes from negative, to zero, to positive.
Point of Inflection
- In the example graph we were working on we didn't get to see it, but you'll often come across parent graphs that will seem to plateau like this
- This is called 'the point of inflection' or where the function changes concavity (which we'll discuss when we get to second derivatives)
- A concave up interval of the parent graph corresponds to a negative interval on the derivative graph and vice versa.
In some instances, we can see that the derivative doesn't cross the x-axis. Instead, it touches it and goes back towards the direction it came in. This would look something like this
We see that the graph doesn't completely cross the x-axis, but it does touch it. This means that this could be a local min/max. If we look at the parent graph of this:
We can see that the graph doesn't have that typical max/min bell curve to it. Instead, it plateaus a bit and then continues decreasing.
- We very very briefly talked about concavity and here we can note that as the parent graph approaches where the derivative touches zero, it is concave up. However, as it passes that point, the parent graph is concave down.
- Therefore, we can state that when the derivative goes from positive, to zero, to positive (+,0,+), OR negative, to zero, to negative (-,0,-), that there could be a point of inflection there.
- Concavity will mostly be talked about when we get to second derivatives but this is also a good way to look out for points of inflection when you're looking at the first derivative.
That's the first derivative and it's graph for you! Now you know what to look for and what they mean contextually. If you have any questions, please let us know!
📌 Here's some great links that might help you if you're still a bit confused!