My Math Forum Are partial derivatives the same as implicit derivatives?

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 November 22nd, 2013, 04:21 PM #1 Newbie   Joined: Nov 2013 Posts: 4 Thanks: 0 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.
 November 22nd, 2013, 05:06 PM #2 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 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, $\dfrac{df}{dx}= \dfrac{\partial f}{\partial x}+ \dfrac{\partial f}{\partial y}\dfrac{dy}{dx}$. Since f(x,y)= 2x+ y, [latex]\dfrac{\partial f}{\partial x}= 2[latex] and $\dfrac{\partial f}{\partial y}= 1$. 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, "$\dfrac{df}{dx}= \dfrac{\partial f}{\partial x}+ \dfrac{\partial f}{\partial y}\dfrac{dy}{dx}$" becomes  which we can solve to get, as before, $\dfrac{dy}{dx}= -2$. So, while we can use "partial derivatives" to find the "implicit derivative", no, they are NOT the "same thing".
 November 23rd, 2013, 11:44 AM #3 Newbie   Joined: Nov 2013 Posts: 4 Thanks: 0 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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