what is explicit and implicit function?

Answer 1

In calculus, implicit differentiation can be applied to implicit functions. This is by an application of the chain rule, to calculate derivatives without necessarily making y an explicit function of x. So, implicit differentiation is nothing more than a special case of the chain rule for derivatives.

Consider for example


This function can be manipulated normally by using algebra to change this equation to an explicit function:


Differentiation then gives . Equally, one can directly differentiate the implicit equation:



Solving for :


An example of an implicit function, for which implicit differentiation might be easier than attempting to use explicit differentiation, is


In order to differentiate this explicitly, one would have to obtain (via algebra)

,
and then differentiate this function. This creates two derivatives, one for y > 0 and another for y < 0. Implicit differentiation avoids this.

One might find it substantially easier to implicitly differentiate the implicit function;


thus,


Sometimes standard explicit differentiation cannot be used, and in order to obtain the derivative, another method such as implicit differentiation must be employed. An example of such a case is the implicit function y3 − y = x. It is impossible to express y explicitly as a function of x (at least using elementary means, although the cubic formula will suffice for restricted values of x and y), which means that cannot be found by explicit differentiation. Using the implicit method, can be expressed;


factoring out shows that

which yields the final answer

Answer 2

explicit functions is something that you can call "in your face" function OR a function which you don't have to solve for.

it tells you simply and plainly what a variable is dependent on

For example y = 5x is an explicit function.

it simply tells you that y is 5 times x

ON the other hand, implicit function is something you have to figure out. you cant just look at it and state what its relation is

for example:

y= 5x -8y

here y is dependent on x AND y, itself

you must add 8y both sides to solve for 9y = 5x; y = 5x/9

That is a very simple example

a complicated implicit function, where you really cannot solve for y is:

y = 5x + sin(y) - 3^(y)

here you cannot solve this function and simply say that

y = blah blah

you must state y -sin(y)+3^(y) = 5x

is that clear?