simple differentiation(urgent)

👨🏻‍🏫 學術台數學

Zidane

2011-05-30 1:06 am

Why is it 2y(dx/dy) but not just simply 2y? I know we want to solve for dy/dx, but I can't understand it.

thanks

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更新1:

Sorry, I can't get what you mean. y^2 is not in terms of x. therefore, d/dx(y^2) = 2y^(2-1) = 2y but why in the next step there is a dy/dx?

更新2:

thanks, I can now understand that. But why can't I change the subject to y = (1-x)^1/2), and then solve it? By doing this, I can only get 0.5(1-x)^(-1/2). Why this method can't be used? thanks

更新3:

I mean is it possible to differentiate all the implicit functions by changing their subject first? (though it may be more difficult and complicated)

回答 (2)

myisland8132

2011-05-30 1:43 am

✔ 最佳答案

This is just the chain rule which said that

du/dx = (du/dy)(dy/dx)

By letting u = y^2,

dy^2/dx

= (du/dx)

= (du/dy)(dy/dx)

= (2y)(dy/dx)

Hope it helps !

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Ho-yin

2011-05-30 7:36 am

change the subject to y = (1-x)^1/2 ...(A)
then differentiate, you should get

0.5 (1-x)^(-1/2) (-1) = -0.5(1-x)^(-1/2)
(Perhaps you are not familiar with chain rule, so you omit the (-1) )

But if you directly differentiate, you get -1/(2y) ...(B)

But if you sub (A) into (B), you'll get the same.

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