sending email outside of network fails

Hello,
I have asp.net page that sends email fine locally.
Mail server (MS exchange server) is located in network and I can send email to my domain email address.
ie. server name: smtp.mysite.net and I can send email to bob@mysite.net

However, I can't send email to Gmail or Yahoo etc, and  get error message as follow

Error Occured, infor=The SMTP server requires a secure connection or the client was not authenticated. The server response was: SMTP authentication is required.

I googled and found code to add, but still not sending.

Dim SmtpUser As New System.Net.NetworkCredential()
SmtpUser.UserName = ""
SmtpUser.Password = ""
SmtpUser.Domain = "mail.mysite.net"

smtpClient.UseDefaultCredentials = False
smtpClient.Credentials = SmtpUser
smtpClient.Host = "mail.mysite.net"
smtpClient.DeliveryMethod = SmtpDeliveryMet ...