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November 22nd, 2013, 04:21 PM   #1
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Are partial derivatives the same as implicit derivatives?

I am about to pull my hair out. Some teachers and books refer to partial derivatives while others refer to implicit differentiation. Are they the same thing? Please explain.
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November 22nd, 2013, 05:06 PM   #2
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Re: Are partial derivatives the same as implicit derivatives

No, they are not at all the same thing. "Some texts and teachers" refer to "implicit differentiation" when they are referring to , the function y as given by an implicit function of x. That is, we are given something like f(x, y)= constant. We could, at least theoretically, solve that equation for y [b]as a function of x. In either case we have y as a function of the single variable, x.

We take partial derivatives of a function of more than one variable. That is, we do NOT have some function of x and y equal to a constant, it is the function of x and y (and z and u and whatever other independent variables we need), it is the function, f, itself we differentiate.

What may be confusing you is that, to find an implicit differentiation, we can use a partial derivative. To take a very simple example, suppose we are given y, defined as a function of x, by f(x,y)= 2x+ y= 7. We could immediately solve that equation for y: y= 7- 2x. Differentiating y with respect to x, dy/dx= -2.

Or we could use the "chain rule" to say that, with f= 2x+ y= 7 a function of x and y and y a function of x, . Since f(x,y)= 2x+ y, [latex]\dfrac{\partial f}{\partial x}= 2[latex] and . But f(x,y)= 2x+y= 7 also means that f(x,y) is the constant, 7, so its derivative, with respect to x, is 0.

That is, "" becomes which we can solve to get, as before, .

So, while we can use "partial derivatives" to find the "implicit derivative", no, they are NOT the "same thing".
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November 23rd, 2013, 11:44 AM   #3
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Re: Are partial derivatives the same as implicit derivatives

Ok, so what do you you to find the point at which a plane is tangent to a sphere?
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