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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 2

Lesson 27: Exponential functions differentiation

# Derivative of 2ˣ (old)

An older video where Sal finds the derivative of 2ˣ using the derivative of eˣ and the chain rule. Created by Sal Khan.

## Want to join the conversation?

• Hm, why did e get brought into this? Why wouldn't the answer be x2^x-1? • Why is the derivative of (ln2)x equal to ln2? Shouldn't it be 1/(2x)? Thanks for your help! •   Basically, the "ln2" in (ln2)x acts as the coefficient in front of x. It would be like taking the derivative of 1x, which would be 1, but instead replace "1" with "ln2".
• At Sal wrote 2 as e^(ln 2) . Please explain . • Will this work with any constant? • At , why did he convert 2^x to (e^ln 2)^x? Why should we write it in that form? • He did it that way to avoid using implicit differentiation. But, you don't have to do it like that. For example:
y=2^x
log₂ y = log₂(2^x)
log₂ y =x
d(log₂ y) = dx
dy/(y ln 2)= dx
Rearrange to:
dy/dx = y ln 2
since y=2^x
dy/dx = (2^x )(ln 2)

Although in this simple case there was not much to be gained by using the fact that f(x) = e^(ln (f(x)), there are problems where trying to solve the derivative any other way is nightmarishly difficult. So, it is a good method to learn.
• what is e and where can i find a video about it? • Around I get confused when Sal starts using the Lebowitz notation in a way that is not '''d/dx". Can someone explain what is happening or refer me to some material that will allow me to master the Lebowitz notation, because that seems to be significantly hampering my understanding of Derivates/Calculus. • Watch the two preceding videos on the chain rule (Chain rule intro & Chain rule definition) to understand a less confusing example of it's application. Here is what is going on at in a notation you might be more comfortable with:
f(x)=2^x and 2^x=(e^ln2)^x
therefore f(x)=(e^ln2)^x
d f(x)/d((ln2)x)=e^((ln2)x)) In plain language this is saying "the derivative of the function f(x) with respect to the natural log of 2 times x"
(1 vote)
• Are logarithms, the number e and trigonometric functions related to one another in some way?
If so then how? • Logarithms, e, and trigonometric functions are all related to some extent with e and trigonometric functions having the strongest relationship. Using Euler's Formula that e^ix = cos x + i(sin x) (or cis x pronounced like an osculation) we can see that cos x = (e^ix + e^-ix)/2 and that sin x = (e^ix - e^-ix)/2i. A weaker link between logarithms and e is that it is common to use logs with base e, i.e., ln, i.e., the natural log. I hope that this has been informative and if you don't know what osculate means you should look it up because it is a fun word.
• I'm confused about why the chain rule has to be used at all after the expression is rewritten at about into the form e^x. I thought the derivative of e^x is exactly just e^x? Yet at about , Sal goes on to multiply e^x by the derivative of the x. Is it that we have to do that if the x on the e^x isn't just a simple x, to put it crudely, but is written as a specific operation like (ln 2)x? Thanks in advance.  