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Differentiation


paulo1913
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What is differentiation used for?

Differentiation is used to find the rate of change. For example, velocity is the time rate of change of displacement; acceleration is the time rate of change of velocity. So, velocity is the first derivative with respect to time of displacement and acceleration is the first derivative with respect to time of velocity(since acceleration is the derivative of a derivative, you can say acceleration is the second derivative with respect to time of displacement). If you've taken a high school level physics course, you've probably used the kinematics equations(equations describing motion) which take advantage of this.

 

If you have a graph, it is the slope of the graph.

 

For finding out how where the values of a function are heading to.

 

That seems closer to the concept of limits.

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Isn't it bassically just the way something varies? So differentation would be used to find differences in everything, but since your talking math. I would say you could use it for Exponential decay and growth. Me personally I like to use it with the reaction rate of enzymes.

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Differentiation and calculus in general must be the most widely applied section of mathematics. I think it has applications in just about all branches/applications of mathematics.

 

So what is it used for? As has been stated, differentiation gives you the infinitesimal change in something with respect to some variable.

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The derivative essentially makes precise the notion of "rate of change" at a particular instant. It is very easy to define "average rate of change" but that necessarily involves a change in the independent variable. You don't have any change in the independent variable at a particular instant!

 

Consider the problem of planetary motion. Newton, and many others, believed there was a force on planets that depended on their distance from the son. But imagine that you are in a rocket ship, above the plane of the planets and you take a "snapshot" of the planets. You could measure the distance from the sun to each planet and (if you knew how force depended on that distance) calculate the force at that instant. Then, since F= ma, you could calculate the acceleration of the planet at that instant. But does that make sense? If acceleration involves "change in velocity over a change in time" and even velocity involve "change in position over a change in time" and there is no "change in time", what possible sense could it make?

 

Newton (and Leibniz) developed calculus, and the derivative, specifically to be able to make sense of that.

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Newton (and Leibniz) developed calculus, and the derivative, specifically to be able to make sense of that.

 

Completely off topic:

I've always preferred the Leibniz notation for derivatives. Which notation do you use more often? I don't even know how to write partial derivatives in the Newton notation.

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