# Why is the chain rule used

## How do I apply the chain rule?

### Derivation with the chain rule: application

The chain rule is one **important rule**with the help of which you can derive complex $ _ "$ chained $" $ functions. The chain rule occurs primarily in combination with other rules, such as the factor rule or the sum rule. An example of the application of the chain rule is this function:

$ \ large {f (x) = (x ^ 2) ^ 3} $

*We want to demonstrate the chain rule, so we don't add up the exponents.* First we have to define the two functional parts u (x) and v (x), because the chain rule says that:

$ f (x) = \ textcolor {green} {u (} \ textcolor {blue} {v (x)} \ textcolor {green} {)} $

In our example this is **Functional part** $ \ textcolor {blue} {x ^ 2} $ the part $ \ textcolor {blue} {v (x)} $ and the **Functional part** $ \ textcolor {green} {() ^ 3} $ the part $ \ textcolor {green} {u (x)} $, so:

$ f (x) = \ textcolor {green} {(} \ textcolor {blue} {x ^ 2} \ textcolor {green} {) ^ 3} $

Now let's look at them again **Derivation formula** $ f '(x) = \ textcolor {green} {u' (} \ textcolor {blue} {v (x) \ textcolor {green} {)} \ cdot v '(x)} $. So we have to do the first **Derivatives** of the individual functional parts and then to calculate them **Derivative function** put together.

So we form them **Derivatives** the individual **Functional parts**:

- $ \ textcolor {blue} {v '(x) = 2x} $, v (x) is also called
**inner function**designated. - $ \ textcolor {green} {u '(x) = 3 \ cdot () ^ 2} $, u (x) is also called
**external function**designated.

After this **Put together** we receive:

$ f '(x) = 3 \ cdot (x ^ 2) ^ 2 \ cdot 2x $

We can simplify this term and then get:

$ f '(x) = 6 \ cdot x ^ 5 $

*Incidentally, we could have summarized the function first and then derived it with the help of the power rule. So:*

$ {f (x) = (x ^ 2) ^ 3} ~~~ \ rightarrow ~~~ {f (x) = x ^ 6} $

$ {f '(x) = 6 \ cdot x ^ 5} $

### Chain rule: example

Another**Sample task** for the chain rule, in conjunction with the **Sum rule**, is the **function**:

$ g (x) = \ textcolor {green} {(} \ textcolor {blue} {3x-2} \ textcolor {green} {) ^ 8} $

The derivation of the **external function** is: $ \ textcolor {green} {u '(x) = 8 \ cdot () ^ 7} $

The derivation of the **inner function** is: $ \ textcolor {blue} {v '(x) = 3} $

Merged according to the **Chain rule **this gives the **Derivation** the function:

$ g '(x) = \ textcolor {green} {8 \ cdot (} \ textcolor {blue} {3x-2} \ textcolor {green} {) ^ 7} \ cdot 3 $

If we simplify this term, we get: $ g '(x) = 24 \ cdot (3x-2) ^ 7 $

To deepen the topic, take a look at the **Exercises**to chain rules.

- How do I prevent shin splints
- Can I train on Ketodiaet
- Cedar is a universal fragrance
- Tomorrow I should study my exam now
- Swimming can cause sinus infection
- What do you miss about Michael Masiello
- Why Are Yamaha Motorcycle Owners So Loyal
- What is 8 8 7 6 6
- Is young love dangerous
- What is the happiest thing about you
- Why did Rohingya Musalman come to India
- Why is Congressman Nunes still obstructing the judiciary?
- How can we solve this question 6
- Why is the dairy industry failing in the US
- Where in France is this place
- Where are the best places to travel
- How is Osmania University doing
- What is your favorite English novel and why
- Can you be injected with cancer
- Why don't people like wool socks
- Are the water spirits demonic
- Which universities accept SAT II Score
- Should I go back to robotics in school?
- What is IQ a predictor for