Louis La Minh Hoang

Today I continued my work on differentiation, building on everything I’ve studied so far while sharpening my understanding through deeper revision. Even though I already know the basic rules, this session pushed me to apply them more fluently and to connect different ideas together in a more advanced way.

I started by reviewing the fundamental techniques — power rule, product rule, quotient rule, and chain rule — but instead of doing simple textbook examples, I focused on problems that combined several rules at once. This kind of revision forces me to think carefully about the structure of each function, especially when the expression contains nested brackets or multiple layers of composition.

Then I moved into second derivatives again, this time using them to analyse the shape of graphs. I practised identifying turning points, determining their nature, and explaining why a function is concave up or concave down in specific intervals. It wasn’t just mechanical differentiation; it was about interpreting what the derivatives really meant. Working out where the gradient increases or decreases helped make the graph feel alive and logical.

I also revised differentiation in the context of real applications — rate of change questions and optimization problems. These problems required more thought than direct calculations. For example, finding maximum values or minimum surface areas meant setting up the function properly, identifying the correct variable to differentiate, and then proving that the critical point was the correct solution. This part of the revision helped me see how differentiation connects directly to real-world ideas.

Finally, I solved several advanced-style problems involving implicit differentiation and situations where functions were defined in terms of each other. These questions were challenging but satisfying, especially when the final result depended on maintaining careful steps and avoiding small errors.

Overall, today’s session strengthened both my confidence and my technique. Continuing the study and revision of differentiation helped me see how all the rules and concepts support each other, and how each level of understanding builds toward more complex and elegant mathematics.