elementary calc
posted by sarah .
f(x) passes through (0,2). the slope of f at any point is 3 times the ycoordinate. find the derivative at 1

When a derivative of a function is a multiple of the y value of the function, it suggests an exponential function.
let the function be y = a e^(kx), where a and k are constanst
dy/dx = ak e^(kx)
given ak e^(kx) = 3y = 3(a e^(kx))
then ak = 3a
k = 3
so we know y = a e^(3x)
at (0,2)
2 = a e^(0)
a = 2
then y = 2 e^(3x)
then dy/dx = 6 e^(3x)
when x=1
dy/dx = 6 e^3 
dy/dx = 3 y
dy/y = 3 dx
ln y = 3 x + c
y = C e^3x
if x = 0, y = 2
so
y = 2 e^3x
when x = 1
y = 2 e^3 = 40
dy/dx = 3y = 120