Question

Consider the curves in the first quadrant that have equations y=Aexp(6x) where A is a positive constant. Different values of A give different curves. The curves form a family, F. Give a formula g(y) for the slope at (xy) of the member of F that goes through (xy). The formula should not involve A or x.

Answer 1

i am getting g(y)=6y, but i am probably wrong. do you know the answer?

Answer 2

i will write what i did, and you can see if it makes sense. you have to write the slope in terms of y so say (x,y) is on the graph. that means it is
\[(x,Ae^{6x})\] and so
\[x=\frac{1}{6}\ln(\frac{y}{A})\]

Answer 3

then the derivative of
\[Ae^{6x}\] is
\[6Ae^{6x}\] replace x by
\[\frac{1}{6}\ln(\frac{y}{A})\] and get
\[6y\] but i would not bet money on this answer

Answer 4

it was right thanks! :)