Calculus BC instantaneous rate of change

Question

Point 𝑃(𝑥,𝑦) moves in the 𝑥𝑦-plane in such a way that 𝑥=1(2𝑡+1)2 and 𝑦=𝑡𝑙𝑛𝑡 for 𝑡>0. Find the instantaneous rate of change of 𝑦 with respect to 𝑥 when 𝑡=𝑒.

𝐴:-(2𝑒+1)3

B. -(2𝑒+1)32

C. -(2𝑒+1)34

D. -2(2𝑒+1)3

Answer 1

Your notation is somewhat non-standard. Usually we try to collect all coefficients in front of (i.e. multiplied on the left of) any variable expressions, and we denote exponentiation with the carrot " ^ " symbol.

But to answer your question, for a parameterized function ( x(t), y(t) ), the instantaneous rate of change is defined to be the ratio of the derivative of y with respect to t and the derivative of x with respect to t. That is:

dy/dx = (dy/dt) / (dx/dt)

So simply compute dy/dt, compute dx/dt, take their ratio, then plug in t = e to get the desired instantaneous rate of change.