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    I don't understand the bit I circled in the proof. Why as small delta X tends towards 0 does small delta y/small delta X tend toward dy/dx etc. ?


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    (Original post by anoymous1111)
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    I don't understand the bit I circled in the proof. Why as small delta X tends towards 0 does small delta y/small delta X tend toward dy/dx etc. ?


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    The definition of  \frac{dy}{dx} is as the limit of  \frac{\delta y}{\delta x} as  \delta x tends to 0. This is because it is an "instantaneous gradient".
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    (Original post by 16Characters....)
    The definition of  \frac{dy}{dx} is as the limit of  \frac{\delta y}{\delta x} as  \delta x tends to 0. This is because it is an "instantaneous gradient".
    Oh I see. And what exactly does limit mean?


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    (Original post by anoymous1111)
    Oh I see. And what exactly does limit mean?


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    What it is approaching. i.e. the limit of f(x) as x tends to c is the value towards which f(x) approaches as x gets closer and closer to c.
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    (Original post by anoymous1111)
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    I don't understand the bit I circled in the proof. Why as small delta X tends towards 0 does small delta y/small delta X tend toward dy/dx etc. ?


    Posted from TSR Mobile
    Basically, thinking of a graph of y = f(x), to find the average gradient of the graph, you'd do \frac{\Delta y}{\Delta x} = \frac{ f(x+\delta x) - f(x)}{(x+\delta x) - x} = \frac{f(x+\delta x) - f(x)}{\delta x}
    Typically, to find the gradient between 2 points, \delta x is rather big, but if we're trying to find the tangent, we need to make the distance between the two points as small as possible, so we make the tangent at that point to be the gradient of x at the 'limit' as \delta x approaches 0

    If we tried y = x^2
    \frac{dy}{dx} = \displaystyle\lim_{\delta x \rightarrow 0} \frac{\Delta y}{\Delta x} = \displaystyle\lim_{\delta x \rightarrow 0} \frac{(x+\delta x)^2 -x^2}{\delta x} = \displaystyle\lim_{\delta x \rightarrow 0} \frac{2x\delta x + (\delta x)^2}{\delta x} = \displaystyle\lim_{\delta x \rightarrow 0} 2x + \delta x
    As the change in x approaches 0, the gradient approaches 2x

    In the book, instead of saying  f(x+\delta x) it's saying y + \delta y or v + \delta v
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    (Original post by Callum Scott)
    Basically, thinking of a graph of y = f(x), to find the average gradient of the graph, you'd do \frac{\Delta y}{\Delta x} = \frac{ f(x+\delta x) - f(x)}{(x+\delta x) - x} = \frac{f(x+\delta x) - f(x)}{\delta x}
    Typically, to find the gradient between 2 points, \delta x is rather big, but if we're trying to find the tangent, we need to make the distance between the two points as small as possible, so we make the tangent at that point to be the gradient of x at the 'limit' as \delta x approaches 0

    If we tried y = x^2
    \frac{dy}{dx} = \displaystyle\lim_{\delta x \rightarrow 0} \frac{\Delta y}{\Delta x} = \displaystyle\lim_{\delta x \rightarrow 0} \frac{(x+\delta x)^2 -x^2}{\delta x} = \displaystyle\lim_{\delta x \rightarrow 0} \frac{2x\delta x + (\delta x)^2}{\delta x} = \displaystyle\lim_{\delta x \rightarrow 0} 2x + \delta x
    As the change in x approaches 0, the gradient approaches 2x

    In the book, instead of saying  f(x+\delta x) it's saying y + \delta y or v + \delta v
    So as the change in X approaches 0, the gradient of the line approaches the gradient of the tangent? Is that basically what it means?


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    (Original post by anoymous1111)
    So as the change in X approaches 0, the gradient of the line approaches the gradient of the tangent? Is that basically what it means?


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    Yeah!
    I've made this graph on desmos, it should help with the understanding if you still think you need it.

    If you've not used desmos before, you can click on the folder icons on the left to hide and unhide the folders, you can zoom and and stuff, change the equation of the curve to anything you want; it's a pretty useful website!

    the d represents the \delta x, hopefully you can work out the rest, but if not then just ask It should be pretty obvious from the graph that as d approaches zero, the red line [aka the secant] approaches the tangent at the point x_0. I've included a green general curve as well, you should see that as d approaches zero, the green curve approaches \frac{dy}{dx} too!
 
 
 
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