Question

What is the derivative of y=2^x?

Answer 1

ln y = x ln 2
1/y dy/dx = ln 2

dy/dx = y ln 2
or
dy/dx = 2^x ln 2

Answer 2

change y = 2^x
to
ln y = x ln2
now take derivative:
dy/dx / y = ln 2
dy/dx = y ln2
= ln2 (2^x)

Answer 3

y'=2^x*ln2

if y=a^x
y'=a^x *lna

Answer 4

Reiny's solution is cool, using implicit differentiation. You can also do it by noting that

2 = e^(ln 2)
So,

2^x = e^(ln 2 * x)
and now use the chain rule:

d/dx e^u = e^u du/dx, so

d/dx e^(ln 2 * x) = ln2 * e^(ln 2 * x) = ln2 * 2^x