Differentiating Explicit and Implicit Functions

An explicit function is one which is given in terms of
the independent variable.

Take the following function,

y = x² + 3x - 8 

y is the dependent variable and is given in terms of the
independent variable x.
Note that   y is the subject of the formula.

 

 

Implicit functions, on the other hand, are usually given in terms
of both  dependent and independent variables.

 eg:-     y + x² - 3x + 8 = 0          

Sometimes, it is not convenient to express a function explicitly.
For example, the circle  x² + y² = 16 could be written as

  or   

Which version should be taken if the function is to be
differentiated ?

 

It is often easier to differentiate an implicit function without
having to rearrange it, by differentiating each term in turn.
Since y is a function of x, the chain, product
and quotient rules apply !

Example

Differentiate x² + y² = 16 with respect to x.

Compared to

 

Example

Differentiate 2x² + 2xy + 2y² = 16 with respect to x.

 

Example

Find the gradient of the tangent at the point R(1,2)
on the graph of the curve defined by x³+ y²= 5, and determine
whether the curve is concave up or concave down at this point.

Divide through by y

Now substitute to find the particular solution