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 .