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How do I find the derivative of 2^x?

To find the derivative of any number to the power "x", such as 2x, 5x, or even 4.13x, we first consider the general form ax. We need to be a little creative here. We know that any variable y can be rewritten as elny.  If we then say that y = ax then we can say that y= elnax. Note that this is because ln(a)x=xlna. So that means y = ax = elna * x. Now we want to find the derivative of ax, or (elna * x )', which is lna ex(lna). This is because lna is a definite number, and so we derivate this the same way we would e3x (which would be 3e3x). Now, if the derivative equals lna ex(lna) we see that actually, ex(lna) is equal to y, so we can rewrite this further as lnay. Since y = ax we can simplify this finally to lnaax. That means that the derivative of ax is axlna. This is the general form and should be remembered. So, (2x)'= 2xln2. 

Answered by Katerina S. Maths tutor

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