M1二階導數

y = 3^x

lny = xln3

Differentiate both sides with respect to x

d(lny)/dy X dy/dx = ln3 dx/dx

1/y dy/dx = ln3

dy/dx = yln3 = 3^xln3

So, we get d(3^x)/dx = 3^xln3

Differentiate both sides with respect to x again

d^2y/dx^2 = ln3 d(3^x)/dx

d^2y/dx^2 = 3^x(ln3)^2

So, d^2y/dx^2 + 2dy/dx + y

= 3^x(ln3)^2 + 2(3^x)ln3 + 3^x

= 3^x[(ln3)^2 + 2ln3 + 1]

= 3^x(ln3 + 1)^2