f(x) passes through (0,2). the slope of f at any point is 3 times the y-coordinate. find the derivative at 1

Answer 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

Answer 2

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