Product Rule: Derivation Rules & Examples |

Task 1

Form the derivative of the function with .

Theoretically, you could multiply the function out to get around the product rule. For practice, you can apply the product rule directly here.

solution

First identify the functions and .

Form the derivation of each.

If you then use the formula of the product rule and simplify the expression, you get the following solution.

Now you already know the formula of the product rule. But why does the product rule apply at all? Are you interested in the derivation the product rule? Then you will find the answers in the following deepening. Feel free to skip this section if you want to go straight to the examples and tasks.

Try applying this principle directly to a function.

The principle can be continued for any number of other functions that are multiplied with each other.

For a function where any number of other functions are multiplied together, the rule is as follows.

For a function with the derivative is:

Tip:

In order to apply the product rule to any number of functions that are multiplied with each other, you can remember the following: The derivative must be formed from each individual function and this must be multiplied with the other functions.

At the end you can now put the knowledge you have learned to the test and solve the following exercises.

Deriving product rules – examples and tasks

Feel free to write down the formula for the product rule and put it next to you or learn it by heart.

Ask your teacher which formulary you can use.

Product rule – derive root function

A root function can also form a product with another function.

task 3

Calculate the derivative of the function with .

Don’t let the π confuse you. This can be treated like a normal number.

solution

First you can identify the functions and .

In the next step you form the derivation of each. The chain rule is applied to the function.

Put together, the total derivation is as follows.

Product rule – derive e-function

Did you give a product with an e-function? You can also derive this type of function using the product rule.

task 4

Form the derivative of the function with .

solution

Again, first identify the functions and .

Next, you form the derivation of each of them again.

If you now apply the formula of the product rule, you get the following complete derivation.

Product rule – deriving fractions

Any function that is represented as a fraction with can also be written as a product.

Thus, the product rule can also be applied to circumscribed fractions.

task 5

Derive the function using the product rule.

solution

First rewrite the fraction as a product.

In the next step you can identify the functions and .

Next, the derivation of each is formed. The chain rule is applied to the e-function.

If you apply the product rule, you get the following complete derivation.

In the index cards for the product rule you will find more exercises and you can check your knowledge using application and arithmetic tasks!

Product Rule – The Most Important