Derivative Techniques: nx^n-1 and k^x ln k

[bobsmith76]

Homework Statement

Let's take two problems, the derivative of

2^x

and

x^π

For the second one the book says that answer is

πx^π-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

2^x ln 2

why not x2^x-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?

 

[Curious3141]

Homework Statement

Let's take two problems, the derivative of

2^x

and

x^π

For the second one the book says that answer is

πx^π-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

2^x ln 2

why not x2^x-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 2^x, express it as e^xln(2). This is of the form e^kx, where k is a constant.

 

[Deveno]

it makes a BIG difference whether x is "upstairs" (in the exponent), or "downstairs" (being exponentiated).

if you go back to the definition:

d(2^x)/dx = lim_h→0 (2^(x+h)-2^x)/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":

2^x = (e^ln(2))^x = e^ln(2)x

which is of the form e^ax, so has derivative ae^ax, by the chain rule.

 

[Mark44]

2^x 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(2^x) = x2^x-1. Each of the rules has "fine print" that says when it can be applied. Read the fine print.