Question

Implicit and Explicit Functions. How can you get you determine a function that is only implicit and one that is both?

Answer 1

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/implicitdiffdirectory/

Answer 2

When a function is written explicitly, it is mean that it is written in the form:\[\bf y=f(x)\]This means that 'y' is given EXPLICITLY in terms of y = whatever..Some explicit functions are:\[\bf y=2x\]\[\bf y=\ln(x)\]and so on...

As you can see, the explicitly stated function is always has y on one side and the rest of the equation on the other side. Now implicitly stated functions are functions where 'y' cannot be found lying alone as 'y' but instead will be with the rest of the function and will often be in the form \(\bf y^2, y^3..etc\). This means, you no longer have 'y' by itself alone and so it is NOT given explicitly. An example of an IMPLICTLY stated function is::\[\bf x^2+y^2=1\]As you can see, the 'y' is not alone on on side and it is to a power of 2. Hence this is an IMPLICITLY function because 'y' is not given in terms of the rest of the function. Now in this sitution, we can EXPLICITLY state 'y' if we like by rearranging and solving for 'y':\[\bf y^2=1-x^2 \implies y = \pm \sqrt{1-x^2}\]In this case, were able to state 'y' EXPLICITLY by rearranging but we can't always do this. For example:\[\bf \sqrt{x} -x^2+y^2+y^3=7\]In this, we have an IMPLICITLY stated function but we can't change/rearrange it so that 'y' is given explicitly like we were able to in the previous example. In such cases, you must differentiate using Implicit Differentiation which is basically differentiating each term using the Chain Rule with respect to 'x'.

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