if g'(x)=2g(x) and g(-1)=1, then g(x)=

Question

calculus question on derivates

Answer 1

if g'(x)=2g(x) and g(-1)=1, then g(x)=

By definition, g'(x) = dg/dx.

Then,  dg/dx = 2g

Or, dg/g = 2dx

Integrating,

int dg/g = int 2dx

ln(g) = 2x + ln(k)    (where ln(k) is the constant of integration)

g(x) = k.e^(2x)

Since g(-1) = 1, then

1 = k.e^(-2)

k = e^2

So, g(x) = e^(2x+2)