Implicit and Explicit functions

[john3j]

Could somebody please help me understand implicit and explicit functions and derivatives? I was reading from http://en.wikipedia.org/wiki/Implicit_and_explicit_functions
but the jargon that is used it hard for me to understand? Examples of each would be great. I understand that some derivatives of some relations only use implicit derication and some can be taken either implicitly or explicitly. Any help to better understand would be great.

 

[DrPhil]

Could somebody please help me understand implicit and explicit functions and derivatives? I was reading from http://en.wikipedia.org/wiki/Implicit_and_explicit_functions
but the jargon that is used it hard for me to understand? Examples of each would be great. I understand that some derivatives of some relations only use implicit derication and some can be taken either implicitly or explicitly. Any help to better understand would be great.

Lets start with definitions in simple English: explicit means "stated clearly and in detail, leaving no room for confusion or doubt."
If you have toe form y = f(x), that is an explicit function. For instance, the slope-intercept equation for a line is explicit: y = mx + b

Implicit means "implied though not plainly expressed," or in mathematical terms, "not expressed directly in terms of independent variables."
An example is the standard form for a line: Ax + By = C. That example can be made explicit, y = (A/B)x + (C/B). Another example of an implicit form that can be solved explicitly is the Pythagorean Theorem: x^2 + y^2 = r^2

Implicit functions become important when they can NOT be so easily solved for y: x^3 + 6 xy^2 + y^3 = 0 can't be written explicity because of the cross term involving both x and y. That is when implicit differentiation comes to the rescue.

 

[HallsofIvy]

y is an explicity function of x if it is written y= f(x) for some expression f-. That is, we have y by itself on one side of the equation, an expression that involves only x on the other side. In particular, that means that if we are given a value of f, we only have to do the operations indicated in the formula- we are given explicit instructions on how to find the values. Examples would be \(\displaystyle y= x^2\), \(\displaystyle y= sin(x)\), and \(\displaystyle y= 3^x\).

y is an implicit function of x if we are given an equation in which both x and y occur. If we are given a value of x, we could put it into the equation and then have to solve the equation for y- we are implicitely told how to find the values. Examples would be \(\displaystyle y^3- x= 0\), \(\displaystyle sin(xy)= 1/2\), and \(\displaystyle x^2- 2xy+ y^2= 4\).

As for finding the derivative, the basic idea is really the same- with the obvious changes. To differentiate \(\displaystyle y= x^2\) we take the derivative of both sides with respect to x- of course, on the left we just write the derivative "symbol": \(\displaystyle y'= 2x\). If we have an "implicit" function \(\displaystyle x^2- 2xy+ y^2= 4\), differentiating both sides, with respect to x, again writing "y'" for the derivative of y.- \(\displaystyle (x^2)'- (2xy)'+ (y^2)'= (4)'\), \(\displaystyle (2x)- (2y+ 2xy')+ 2yy'= 0\), using the product rule for \(\displaystyle (2xy)'\) and the chain rule for \(\displaystyle (y^2)'\), and then solve for y'. We can write that as \(\displaystyle 2(x- y)+ 2(y+1)y'= 0\) and then \(\displaystyle (y+1)y'= y- x\) so that \(\displaystyle y'= \frac{y- x}{y+ 1}\).