The functions are   and .

Sketch the functions

Let us consider the point passing through .

Now the function is .

Apply derivative on each side with respect to .

                                                          (The power rule : )

Let us consider the point passing through .

Now the function is .

Apply derivative on each side with respect to .

Sum rule and difference rule of derivative  : .

Since the line tangent to both the graphs will have unique slope.

such that  .

Finding the slope by using the two points is .

                 ( Since and )

         ( Since )

The line tangent to both the graphs will have unique slope.

So we equate it to  .

Multiple with on each side

Simplify the equation,

   or 

   or

Let us consider .

 Substitute in , then  .

Substitute in , then .

The point is .

Substitute in , then 

The point is .

Therefore the point are and .

Point - slope form of a line equation is .

So that tangent line goes through points is .

Let us consider .

Substitute in , then .

Substitute in , then .

The point is .

Substitute in , then 

The point is .

Therefore the point are and .

Point - slope form of a line equation is .

So that tangent line goes through points is .

 Graph the tangent line passing through both the functions.

Graph the tangent line passing through both the functions.

 The tangent line equations are and .