Christos
  • Christos
so according to the derivative definition we have lim h->0 f(x+h)-f(x)/h can someone explain to me what is h ?
Mathematics
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chestercat
  • chestercat
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ParthKohli
  • ParthKohli
so do you know how slopes work in the case of simple straight lines? this has a lot to do with the formula slope = \(\Delta y / \Delta x\)
ParthKohli
  • ParthKohli
basically we created derivatives to calculate the slope of a "tangent" at a point. but what exactly is a tangent at a point? look at the series of pictures that are about to follow|dw:1449944455673:dw|?
ParthKohli
  • ParthKohli
|dw:1449944558915:dw|

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Christos
  • Christos
ah right the tangent line needs to touch the function only once
ParthKohli
  • ParthKohli
|dw:1449944605189:dw|
ganeshie8
  • ganeshie8
|dw:1449944653411:dw|
ganeshie8
  • ganeshie8
the tangent line "can" touch the function at multiple places
Christos
  • Christos
wouldn't that be a secant then
Christos
  • Christos
or are you talking just about the extreme cases
ParthKohli
  • ParthKohli
so what we did was this: we considered the point where we had to draw the tangent. how we did it was we considered a point that was soooo close to the original point that they were pretty much indistinguishable by the end, but still distinct when we join the line connecting those two points and apply the formula for slope of a straight line, we get that result\[m = \frac{\Delta y}{\Delta x} = \frac{y_2-y_1}{x_2-x_1}=\frac{f(x+\Delta x)- f(x)}{x+\Delta x - x}\]
Christos
  • Christos
where slope = 0 or approaches inf
Christos
  • Christos
ah i see
Christos
  • Christos
so thats a secant and a tangent at once
Christos
  • Christos
kk good point
ParthKohli
  • ParthKohli
so if it helps you can think of \(h\) as \(\Delta x\) and you're bringing \(\Delta x\to 0\) so that \(x+\Delta x \to x\).
ParthKohli
  • ParthKohli
but we never really actually put \(\Delta x =0\) so that is why it is a limit
ParthKohli
  • ParthKohli
of course we can't do that - because there is a \(\Delta x\) in the denominator
Christos
  • Christos
woah very nice explanation, I think i got the gist of it now
ParthKohli
  • ParthKohli
|dw:1449945126150:dw|
ParthKohli
  • ParthKohli
to support what ganeshie said, yes - the tangent can touch the function at multiple places. the notion of it touching only one point works well for circles though
Christos
  • Christos
so a secant line can be tangent at the same time then
ParthKohli
  • ParthKohli
uh, the points of intersection must be "local" for a secant line if that makes sense.
Christos
  • Christos
i see

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