# Introduction to the product rule

• sapiental
In summary, there are 256 bit strings of length 8, and 64 of those begin with 2 1's. This is determined using the Product Rule.
sapiental

## Homework Statement

a) How many bit strings are there of length 8?

b) How many bit strings are there of length 8 which begins with 2 1's?

Product Rule

## The Attempt at a Solution

a) Since a bit string is either 0 or 1 there are two possibilities for each one. By the rule of products 2^8 = 256 bit strings.

c) the first two choices are fixed so its 11(0/1)(0/1)(0/1)(0/1)(0/1)(0/1)

2^6 = 64 possible different bit strings.

when u allow the preceding 1 to change u basically double the 64 combinations and allowing the first one u quadruple it getting back to the original 256.

Could someone please confirm my result. Danke!

In short: Yes.

## What is the product rule?

The product rule is a formula used to find the derivative of a product of two functions. It states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.

## Why is the product rule important?

The product rule is important because it allows us to find the derivative of more complex functions, such as products of multiple functions. This is useful in many areas of mathematics and science, including physics, engineering, and economics.

## How do you use the product rule?

To use the product rule, you must first identify the two functions that are being multiplied together. Then, you can use the formula to find the derivative of the product. Remember to apply the chain rule if one or both of the functions have a nested structure.

## What is an example of using the product rule?

An example of using the product rule is finding the derivative of f(x) = x^2 * sin(x). Using the product rule, we can find that the derivative is f'(x) = 2x * sin(x) + x^2 * cos(x).

## Are there any special cases for the product rule?

Yes, there are two special cases for the product rule. The first is when one function is a constant, in which case the derivative of the product is simply the constant times the derivative of the other function. The second case is when the two functions are reciprocals of each other, in which case the derivative of the product is always equal to -1.

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