y=2^x

lny = ln 2^x = x* ln2

d(lny)/dy = d(x*ln2) / dy

1/y = ln2 * dx/dy

dy/dx = ln2 * y = ln2 * 2^x
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y= (2^x) / ln2 = ( 1/ln2 ) * ( 2^x)

dy/dx =[d(1/ln2) ]* 2^x + (1/ln2) *[d(2^x)]

=(1/ln2) * ln2 * 2^x

=2^x

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better to understand what you are doing, not just memorizing...