How do you differentiate 2^x?

We can differentiate this implicitly by writing the question as:

y = 2

Then we take the log of both sides:

ln(y) = ln(2x)

Using the rules of logartithms this can be written as:

ln(y) = x ln(2)

Now we can differentiate this easily:

y-1 dy/dx = ln(2)

We can now re-arrange to get:

dy/dx = y ln(2)

And finally we can substitute y to get our answer:

dy/dx = 2ln(2)

So we have shown that the derrivative of 2x is simply 2x multiplied by ln(2)

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