A few days after the battle, Septima hunched over the table and frowned in concentration over her calculus 1 homework on the chain rule. That had been today's lesson and now she was tackling the mountain of assignments about the subject.
''So for sin(2x), I first take the derivative of sin(x) and then that of 2x, right? Then I multiply the two derivatives of each respective term in order to get 2cos(2x), Septima said, almost to herself but accidently verbalized her thought out loud.
''Yes, that is correct,'' Hermione remarked from accross the table.
"Ah, thank you,'' Septima replied with a smile.
Hermione had been Septima's student while a Hogwarts student until Cruciatus-induced torture and a head injury left Septima having to relearn math and arithmancy. She had made her way through high school level algebra in the months since the injury and now was taking a correspondence calculus class at a Muggle university.
Septima had remembered that Hermione completed a Muggle degree in math via correspondence while at Hogwarts, so Septima had decided to enroll in the same program with the help of Hermione.
Pr. Tremaine, a math professor from the Muggle university, had been shocked and sympathetic when Hermione wrote to him and mentioned how her friend Septima Vector (Vector, what an interesting surname, Pr. Tremaine had thought. Almost a magical coincidence.) had been in a serious car accident (Muggle cover story) and suffered a head injury that left her having to relearn math. So, Pr. Tremaine helped Septima figure out a degree plan and she enrolled as a student (Hermione was paying for the degree with income from her magical Archimedes jewelry business).
''I remember the first time that I met you in that hotel room in Venice, Italy when you got your Hogwarts letter and wanted to enroll in arithmancy as a first year. You were working on the chain rule yourself and I tested you on it. Now I guess the rolls have reversed,'' Septima said to Hermione.
''Yes, I recall that,'' Hermione replied, ''but don't worry, Septima. One day in a few years from now you have will your PhD in math.''
Septima smiled at Hermione's words of encouragement. It had been a long and challenging journey for her, but she was determined to overcome the obstacles and achieve her goals.
"I appreciate your faith in me, Hermione," Septima replied. "I must admit, there are moments when I doubt myself, but I'm determined to push through. I've come a long way since that hotel room in Venice, and I won't let this setback define me."
Hermione nodded, her expression filled with admiration. "I've seen your determination and resilience firsthand, Septima. I have no doubt that you will succeed in your studies and earn your PhD. You've already accomplished so much, and I believe in your ability to overcome any challenge."
Septima took a deep breath, her focus returning to the calculus problems in front of her. "Well, then I better get back to work and conquer this chain rule. I won't let it defeat me."Hermione smiled and offered her support. "You've got this, Septima. Remember, if you ever need any help or have any questions, I'm here for you. We'll tackle these mathematical mountains together."
With renewed determination, Septima resumed her studies, knowing that she had a friend by her side who believed in her abilities. As she delved into the complexities of calculus, she couldn't help but feel a surge of excitement for the future that awaited her—a future where she would proudly hold a PhD in mathematics.