Interpretations of derivatives.

Example:   Find out the equation of the tangent line to

                                      x² + y ²  =9

at the point (2, √5 ) .

Solution

We have to be given both the x & the y values of the point.  Notice that this point does lie on the graph of the circle (you can verify by plugging the points into the equation) and thus it's okay to talk regarding the tangent line at this point.

Recall that to write the tangent line we required is the slope of the tangent line and it is nothing more than the derivative evaluated at the specified point. 

Then the tangent line is.

                                      y =     √5 - 2/ √5  ( x - 2)

Now, let's work on some more examples. 

There is a simple way to remember how to do the chain rule in these problems.  Really the chain rule tells us to differentiate the function as usually we would, except we have to add on a derivative of the inside function.  In implicit differentiation it means that every time we are differentiating a term along y in it the inside function is the y and we will have to add a  y′ onto the term as that will be the derivative of the inside function. Let's see a couple of examples.