The TI89 has a perfectly nice builtin tangentline function accessed through the Menu on the graph screen. Suppose that we want to find an equation of the tangent line to the graph of y = 3x^{2}  ln x at the point (1, 3) (This is problem #67 on p. 319 of the Larsen text.) A graph of this function is shown at right using the window . 

Select "A: Tangent" from the menu. 

You can either use the arrow keys to select a point on the graph, then press , or type an xcoordinate and press in answer to the "Tangent at?" question. 

The equation of the tangent line is given as "y = 5.x  2.". The decimal points indicate that this is an approximate answer (although in this case it happens to be exact). 

To get an exact answer to the question "What is an equation of the tangent line..." you can enter a new function. It is:
Define tanline(f, x, c) = (d(f, x)  x = c)*(x  c) + (f  x = c)
The screen shot at right shows the result of using the tanline() function to solve the same problem as above. An advantage of this function over the "A: Tangent" method is that the userdefined function can produce exact results. 
