The functions are  and .

(a)

Consider .

Apply derivative on each side with respect to .

Graph :

Graph the function  and .

(b)

Consider .

Apply derivative on each side with respect to .

Graph :

Graph the function  and .

(c)

Observe the graphs of function and its derivative.

The graph of  and  follow similar pattern.

The graph of  and  are similar except the multiplication factor.

Therefore the derivative of the polynomial is one degree less than the original function.

Hence if the function is  then the derivative of the function is  where .

(d)

The function is .

Find .

Limit definition of derivatives : .

.

Derivative of  is .

Conjecture :

If the function is  then the derivative of the function is  where .

Now the function is  then derivative is .

Hence the conjecture is true.

This is not the proof but its an example to show that the conjecture is true.

(a) 

Graph of the function  and  is

(b)

Graph of the function  and  is

(c) If the function is  then the derivative of the function is  where .

(d) Derivative of  is .