What do we mean by slope especially in application of derivatives? Please explain it in basic way. I want to understand it to apply. When is it to be used?
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In the first place "slope" is a quantity associated to non-vertical lines in an $(x,y)$ coordinate system, see Constantine's answer. The derivative of $f$ at $x_0$ is equal to the slope of the tangent to the graph of $f$ at $\bigl(x_0,f(x_0)\bigr)$. – Christian Blatter Aug 03 '13 at 15:42
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The "slope" of a line is usually introduced in algebra classes, before one ever studies calculus. For example, slope is "rise over run" and if you write the equation of a line in the form $y = mx + b$, then it turns out $m$ is the line's slope. I couldn't tell from your question if you already understand what the slope of a line is. – littleO Aug 03 '13 at 18:53
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In a basic terms, consider some function $ f(x) $ and a tangent line to the curve of the graph of this function at $ x_0 $, now consider a triangle with vertices at $(0,0)$, $(x_0,0)$ and $ (x_0,f (x_0)) $. Slope is a rise over run, or $\frac{f (x_0)}{x_0}$, which is by definition $\tan \theta $, where $\theta $ is the angle tangent line makes with the $ x $-axis, which is, in turn, the same as the derivative of $ f (x) $ at a point $ x_0 $.
Constantine
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Well, in the normal 2-d setting which hopefully is the "basic" setting you are looking for, the derivative $\frac{dy}{dx}$ is the gradient of the equation $f(x,y)=0$.
So as an example, the line $y=mx+c$ has the gradient $m$ and $y$-intercept $c$. When you differentiate, you get $\frac{dy}{dx}=m$. It is normally easier to find the derivative than to compute the gradient at a point, which is why we use derivatives.
I hope this is what you are looking for.
BlackAdder
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yeah i understood mathematically , but when and how do we apply it in physics – gksingh Jul 08 '13 at 15:23
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1For example, if you did an experiment and you plotted your data points on a graph, then for example, if you have distance against time, the gradient would be the same as the derivative which is the velocity at that point. – BlackAdder Jul 08 '13 at 16:08
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You can approach this in a Geometric perspective. So, the first you can put in the table is:
Trying with some function in the graph is $f(x)=x^2$ in green, you can see the parabola in the image. And then we need to understand well what is the slope. for doing that, in order to be more clear for you I use a similar function $f(x)=0.088x^2$ in red. (just because is more clear the slope). In other terms the slope is the tangent. Usually the tangent is comprehended like the line that only have one point in a function but this is not exact, some times you have messy curves. So for trying to understand the slope $(m)$ or the tangent we need to use two different points. $P$ and $Q$. The line on orange is not a tangent the line in blue it is. So if you approach the point $Q$ to $P$ you have the tangent in $P$. As you go near and near to $P$ (moving $Q$) you approach the slope. And that is the tangent line of $f(x)=0.088x^2$ on $P$.
Understanding in this way means that the slope is some kind of "rate" meaning the instantaneous rate on $P$ of the change on $f(x)$. Try to remember "this for that" for example miles per hour, gallons per minute, kilowatts per hour, etc. Is the rate in the very "moment" (this is not exact is just and interpretation in terms of time).
If you need more examples, you need to think and look around you carefully. In physic terms, you need to see if different events are related. Is that is so, then you need to think how this events are related. A physical phenomena appears in front of you just because you can see something changing in relation to another thing(s). In the examples before, you can see something moving related to some fix point, this rate is named velocity (miles per hour). If you need to know the velocity that you need to fill a tank you are thinking how much liquid i needed to fill the space you have for it in terms of time (gallons per minute). The same happens in terms of the power you use in time, for example running a power plant or the waste of your laptop battery. All this are Physical uses of the derivative. We can also say that is common used for "optimization processes". So, if you are able to know what happen in the instant relation of at least two variables, you can compute which of all the possibilities are better based on some expectation you have for the behavior of that system. See one example. You need to travel to some place with some money and some hurry. Knowing the relation between waste of gas and time, you can make a better plan for optimizing your money expenses and be as soon as possible to your destiny.
Haizum Skallah
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