<PAGE>
 
                   AMENDMENT TO EXECUTIVE INCENTIVE AGREEMENT


     This Amendment, effective as of the 16th day of December, 1997, by and
between Cox Communications, Inc. a Delaware corporation ("Cox"), and Ajit M.
Dalvi, Senior Vice President of Programming & Strategy of Cox (the "Employee").

                              W I T N E S S E T H:


     WHEREAS, Employee has participated in an appreciation program, pursuant to
which Employee is entitled to the payment of compensation based on the
appreciation in the value of Discovery Communications, Inc. ("DCI");

     WHEREAS, the terms of the appreciation program were set forth in a letter
agreement Employee received from Cox Cable Communications, the corporate
predecessor to the businesses of Cox, which was dated February 24, 1993 (the
"Agreement"); and

     WHEREAS,  pursuant to the Agreement, Dalvi is entitled to receive a payment
in the amount of $1,000,000, payable under the terms of the Agreement;

     WHEREAS, Cox and Employee intend to negotiate modifications to the terms of
the Agreement in accordance with the terms of this Amendment, and have concluded
the execution of this Amendment is in the best interest of the Corporation.

     NOW, THEREFORE, the parties hereto agree to amend the Agreement as follows:

                                       1.

     Employee and Cox agree that $500, 000 of the payment otherwise payable to
Employee under the Agreement shall be reinvested in a phantom equity interest in
DCI, and will adjust in value in the same fashion as set forth in the Agreement
throughout the period ending on December 31, 1999.  The maximum value that may
be payable to Employee under this Section 1 of the Amendment shall be
$1,500,000.  This phantom equity interest will be payable to Employee as soon as
practicable after the later of January 1, 2000 or the date of Employee's
termination of employment with Cox, subject to earlier payment to the extent
provided for in the Agreement.


 

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                                       2.

     Employee and Cox agree that the remaining $500,000 of the payment otherwise
payable to Employee under the Agreement shall be paid to him in the form of
13,675 shares of Class A common stock of Cox.  The shares of Class A common
stock issued to Employee hereunder shall be subject to forfeiture by Employee in
the event he does not remain in the continuous employment of Cox or one of its
affiliates from the date hereof until January 1, 2000.  As  of such date, the
risk of forfeiture shall lapse.


                                       3.

     All capitalized terms used in this Amendment shall have the same meanings
ascribed thereto in the Agreement.

     IN WITNESS WHEREOF, the parties hereto have executed or caused this
Amendment to be executed, on the date first above written.

                         COX COMMUNICATIONS, INC.



                         By:    /s/ James O. Robbins 
                                --------------------               
                         Title: President/CEO
                                --------------------



 

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