How derivatives used to find out rates of change?

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Wasee

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Wasee asked the Naked Scientists:

Hello,

I know what derivatives are and how to find the derivative of a function, but what bothers me is how they are used to find out rates of change velocity, accelaration etc

Thanks

What do you think?

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Offline Chemistry4me

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How derivatives used to find out rates of change?
« Reply #1 on: 12/01/2009 09:34:56 »
distance (differentiate) > velocity (differentiate) > acceleration
Draw the graph and then its derivative and you'll see it as the gradient function. The x-intercepts of the gradient function are the turning point(s) of the original graph.

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Offline Chemistry4me

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How derivatives used to find out rates of change?
« Reply #2 on: 12/01/2009 10:14:11 »
If I was to draw a diagram, like so:
[diagram=391_0]

The co-ordinates of P are (x, (f(x)). If the x-coordinate of Q is (x+h) then the y-coordinate of Q is f(x+h). Gradient of PQ = QR/PR
                          = (f(x+h) - f(x))/h

As Q 'slides down' the curve: In the limit, as Q→P, and h→0, the gradient of the chord PQ approaches the gradient of the tangent at point P.
Gradient of the tangent at P = lim(h→0) (f(x+h)-f(x))/h
This is often referred to as the gradient function written f'(x)
i.e f'(x) = lim(h→0) (f(x+h)-f(x))/h
This limit is used when calculating the derivative of f(x) from first principles. The derivative f'(x) is interpreted as the rate of change of y with respect to x. Hope you understood that [:)]

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lyner

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How derivatives used to find out rates of change?
« Reply #3 on: 12/01/2009 12:09:13 »
Mr Worthington told us that in 1961!!!
And it hasn't changed.

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Offline Chemistry4me

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How derivatives used to find out rates of change?
« Reply #4 on: 12/01/2009 22:41:30 »
Mr Worthington told us that in 1961!!!
I presume that was your maths teacher.
Am I correct (with the calculus bit)?

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lyner

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How derivatives used to find out rates of change?
« Reply #5 on: 12/01/2009 23:00:20 »
A legend!!