Gonna Make Leibniz Proud With This One [Korean CSAT 2020]

0:00 Introduction to the problem (twice actually, to help you understand).
1:33 Solution: Common part (for methods 1 and 2).
3:56 Method 1 to find da/dt. Implicit differentiation on Eq. (1).
7:53 Method 2 to find da/dt. Solve for a and differentiate.
9:46 Method 2-1. 1st method to eliminate k.
11:26 Method 2-2. 2nd method to eliminate k.
A very hard calculus problem where you have to use implicit differentiation or chain rule, or both. The most irritating thing about this problem is that there exist so many variables (x, t, a, and k, the last one introduced by me), so when taking derivatives, you must be well aware of which variable is being constant (fixed) and which variable is changing. And you must be well aware of which variable is dependent on which (here, variables a and k change as t changes, meaning that they are both functions of t).
In order to make the solution less confusing, I used Leibniz notation for derivatives (dy/dx, da/dt, etc.) instead of Lagrange's notation (y', f'(t), etc.). For this problem in particular, the Leibniz notation helps you more intuitively distinguish the derivative with respect to x (d/dx) and the derivative with respect to t (d/dt).
South Korean College Scholastic Ability Test (CSAT) is a nationwide test taken by South Korean high school seniors in order to apply for the college. Check other problems at the following list:
• Korean CSAT Problems
#KoreanCSAT #calculus #differentiation #tangent #derivatives #ChainRule #ImplicitDifferentiation #suneung #CSAT2020 #KillerProblem
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