Derivative of 2^x

  • Thread starter bobsmith76
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  • #1
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Homework Statement



Let's take two problems, the derivative of

2x

and

xπ

For the second one the book says that answer is

πxπ-1

Well if you can do that for the above, then why not for the first problem?

The book gives the answer to the first problem as

2x ln 2

why not x2x-1?

essentially what i need to know for finding derivatives, there are two techniques.
1. nx^n-1
2. k^x ln k

when do I use which technique?
 

Answers and Replies

  • #2
Curious3141
Homework Helper
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Homework Statement



Let's take two problems, the derivative of

2x

and

xπ

For the second one the book says that answer is

πxπ-1

Well if you can do that for the above, then why not for the first problem?

The book gives the answer to the first problem as

2x ln 2

why not x2x-1?

essentially what i need to know for finding derivatives, there are two techniques.
1. nx^n-1
2. k^x ln k

when do I use which technique?

Because that rule only works when the power of the x term is a constant (independent of x).

To differentiate 2x, express it as exln(2). This is of the form ekx, where k is a constant.
 
  • #3
Deveno
Science Advisor
908
6
it makes a BIG difference whether x is "upstairs" (in the exponent), or "downstairs" (being exponentiated).

if you go back to the definition:

d(2x)/dx = limh→0 (2(x+h)-2x)/h

you can see that you're not going to get an easy way to simplify.

basically, e is the "natural base" for exponential functions, and other bases have logarithms as a "conversion factor":

2x = (eln(2))x = eln(2)x

which is of the form eax, so has derivative aeax, by the chain rule.
 
  • #4
35,285
7,128
2x and xn are very different types of functions. The first is an exponential function, so called because the variable is in the exponent. The second is a power function, so called because the variable is in the base, which is raised to a fixed power.

There is no single differentiation rule, other than the definition, that covers both of these functions. You are misusing the power rule to conclude that d/dx(2x) = x2x-1. Each of the rules has "fine print" that says when it can be applied. Read the fine print.
 

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