The formula for the definition of a derivative is long, and looks confusing at first sight — but in reality, it’s pretty simple. In fact, it’s as simple as calculating the slope of a line. Using the formula may seem like overkill — especially when you’re finding the derivative of a simple function using the power rule — but it’s especially useful for more complex functions in which finding the derivate is hard.
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Math Journal September 21, 2009.
I found this video the be very illustrative with the actual writing out of the equation as it is explained by the speaker. It is a nice and slow explanation that touches on every little detail there is to know. There are no skipped steps or shortcuts so it is easy to follow along and understand what exactly is happening. And instead of just showing me the shortcut i can come to the understanding of how that shortcut was derived in the first place, so that when i use the shortcut in the future it will not just be a blind plug and chug problem. Overall it is a great illustration of finding the derivative of an equation.
Math Journal September 23, 2009.
This video provides an excellent explanation of how to find the derivative by using the definition of the derivative. It goes at a pace that makes it really clear. Each step is explained clearly and thoroughly. It doesn't skip any steps, even the small ones so that you can develop an understanding of what this explanation is actually trying to explain instead of just knowing how. For a simple explanation of how to get the derivative from the definition this video is great. This example makes this concept seem very easy. I think it would be beneficial if there was a more complicated example that deals with square roots, fractions, and or some trig.
This video is a good way of showing how to explain what a derivate is however there is a much simpler way to find a derivative in calculus on a regular basis. The formula takes up way too much paper (arent we going green???) and too much time to even really use it for multiple equations. The power rule is a much more time efficent and simpler way of finding a derivative. Also in general you will probably make more mistakes using the formula then by using the power rule. Yet the essence of knowing where the derivative comes from offers more insight into the basis of calculus. I guess it just depends on what way you look at it. On one hand the power rule is probably best for tests or time situations. Yet this definition of a derivative is most likely better for grasping a true understanding of the actual math behind it.
I found this pencast to be a good review. It really does touch on every step of finding a derivative of an equation, is easy to follow and doesn't drag! This pencast goes over using Newton's Quotient vs. using the power rule and how the power rule is a good double check on your answer but shows the importance of understanding why the power rule works (derivation).