Derivative of 2ˣ (old)

let's see if we can take the derivative with respect to X of 2 to the X 2 to the X power and you might say hold on a second we know how to take the derivative of e to the X but what about a base like 2 we don't know what to do with 2 and the key here is is to write rewrite 2 to the X so that we essentially have it as e to some power and the key there is to rewrite 2 so how can we rewrite 2 so it is e to some power well let's think about what e let's think about what e to the natural log of 2 power is the natural log of 2 is the power that I would have to raise e to to get to 2 so if we actually raise e to that power we are going to get to 2 so what we could do instead of writing 2 to the X we could rewrite this as e we could rewrite 2 as e to the natural log of 2 e to the natural log of 2 and then raise that to the X power so this is the X power and yellow this is the X power in yellow and so let's do that right over here so instead of taking the derivative with respect to X of 2 to the X let's say let's just take the derivative with respect to X of it the exact same expression rewritten of e of e to the natural log of 2 e to the natural log of 2 raised to the X power raised to the X power let me put this X in that same color DX now we know from our exponent properties if we raise something to some power and then raise that to another power we can take the product of the two powers let me rewrite this just remember if I have a to the B and then I raise that to the C power this is the exact same thing as a to the B times C power so we can utilize this exponent that exponent property right here to rewrite this as being equal to the derivative with respect to X of e e to the natural log of 2 times X e to the natural log of 2 times X times X and what's neat about this is now we got this into a form of e to the something so we can essentially use the chain rule to evaluate this so this derivative is going to be equal to the derivative of e to the something with respect to that something well the derivative of e to the something with respect to that something is just e to that something so it's going to be equal to e to the natural log of 2 to the natural log of 2 times X so let me make it clear what I just did here this right over here is the derivative of e to the natural log of 2 times X with respect to with respect to the natural log of 2 let me make it a little bit clear with respect to the natural log of 2 times X so we took the derivative of e to the something with respect to that something that's this right here it's just e to that something and then we're going to multiply that by and this is just an application of the chain rule of the derivative of that something with respect to X so the derivative of natural log of 2 times X with respect to X is just going to be natural log of 2 this is just going to be natural log of 2 the derivative of a times X is just going to be equal to a this is just the coefficient on the X and just to be clear this is the derivative of natural log of 2 times X with respect to X so we're essentially done but we can simplify this even further this thing right over here can be rewritten and let me draw a line here just to make clear that this equal sign is a continuation from what we did up there but this e to the natural log of 2x we can rewrite that using this exact same exponent property as e to the natural log of 2 to the natural log of 2 and then all of that raised raised to the X power raised to the X power and of course we're multiplying it times the natural log of 2 so times the natural log of 2 well what is e to the natural log of 2 well we already figured that out that is exactly equal to 2 this right over here is equal to two and so now we can simplify this whole thing the derivative of 2 to the X is equal to and I'll switch the order a little bit it is the natural log of 2 that's this part right over here times 2 to the X times 2 to the X or we could write it as 2 to the x times the natural log of 2