# Solving the Chain Rule: A Visual Guide

• QuarkCharmer
In summary: I hope the identity is obvious! :smile:I meant differentiate one of the sides. :wink:I'll get to it eventually. I'm not in the mood for playing around with trig identities right now. :tongue2:In summary, using the product and chain rule, the derivative of (x+(x+sin^2(x))^3)^4 is 4(x+(x+sin^2(x))^3)^3(1+3(x+sin^2(x))^2)(1+2sin(x)cos(x)).
QuarkCharmer

## Homework Statement

$$\frac{d}{dx}(x+(x+sin^2(x))^3)^4$$

## Homework Equations

Calc up to Chain Rule.

## The Attempt at a Solution

Using product and chain rule I got:
$$\frac{dy}{dx}=4(x+(x+sin^2(x))^3)^3(1+3(x+sin^2(x))^2)(1+\frac{d}{dx}sin^2(x))$$

Then I calculated the derivative of sin^2(x):
$$\frac{d}{dx}sin^2(x)=2sin(x)cos(x)$$

and put that into the derivative to get:

$$y'=4(x+(x+sin^2(x))^3)^3(1+3(x+sin^2(x))^2)(1+2sin(x)cos(x))$$

Do I further simplify this? It does not seem obvious to me. What should I be paying attention to next?
Here is another problem I worked out. I think it's correct, assuming I am using the chain-rule correctly?
[PLAIN]http://img824.imageshack.us/img824/2691/imag0025ed.jpg
It simplified down to something sort of pretty, but that other one looks horrible!

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That is correct. I think you can't simplify it further. But at entry level, if you're afraid of using chain rule for really lengthy functions, like this, you can substitute a suitable part of it. ie.

$$\frac{d}{dx}(x+(x+sin^2(x))^3)^4$$
let,
$$(x+sin^2(x)) = z$$
so,
$$\frac{d}{dx}(x+(x+sin^2(x))^3)^4 = \frac{d}{dx}(x+z^3)^4$$

Now, do differentiate in the same way. The final answer will be same, but it may decrease the chance to make a mistake.

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That is basically what I was doing. Only I did it in my head.

QuarkCharmer said:
That is basically what I was doing. Only I did it in my head.

Hi QuarkCharmer!

I think you have a bracket in the wrong place, but otherwise it's fine.

In your second one, if you're wondering why it's so neat, that's because you could rewrite the bottom as (√2)cos(πx - π/4) …

then the top becomes cossin - sincos = sin((πx - π/4) - πx) = sin(-π/4) = -1/√2.

tiny-tim said:
Hi QuarkCharmer!

I think you have a bracket in the wrong place, but otherwise it's fine.

In your second one, if you're wondering why it's so neat, that's because you could rewrite the bottom as (√2)cos(πx - π/4) …

then the top becomes cossin - sincos = sin((πx - π/4) - πx) = sin(-π/4) = -1/√2.

I see what you did there, but unless it's really obvious, those phase shifting identities would be going a bit too far I would think?

Hi QuarkCharmer!
QuarkCharmer said:
… phase shifting identities …

You make it sound like something out of Star Trek!

Surely it's always simpler to deal with one cos rather than a sum of two?

(You might like to try differentiating a general cos(x+A)/cosx, = cosA + sinAtanx )

tiny-tim said:
Hi QuarkCharmer!

You make it sound like something out of Star Trek!

Surely it's always simpler to deal with one cos rather than a sum of two?

(You might like to try differentiating a general cos(x+A)/cosx, = cosA + sinAtanx )

Hey, I didn't make up the nomenclature! I learned about alternating current before I knew what trigonometry was, so I just think of it that way.

I'll give that a shot, but do you mean to prove the equality, or differentiate one of the sides?

I hope the identity is obvious!

I meant differentiate one of the sides.

## 1. What is the chain rule in mathematics?

The chain rule is a fundamental rule in calculus that allows us to find the derivative of a composite function. It states that the derivative of a function composed with another function is equal to the product of the derivative of the outer function and the derivative of the inner function.

## 2. Why is the chain rule important?

The chain rule is important because it allows us to find the rate of change of complex functions that are composed of multiple simpler functions. It is essential in many applications of calculus, such as optimization, physics, and engineering.

## 3. How do you solve the chain rule visually?

Solving the chain rule visually involves breaking down a composite function into its individual functions and their derivatives, and then combining them using a "chain" of arrows to represent the composition. This can help to simplify the process and make it easier to understand.

## 4. Can the chain rule be applied to functions with more than two components?

Yes, the chain rule can be applied to functions with any number of components. The key is to break down the composite function into its individual functions and their derivatives, and then use the chain rule to find the derivative of each function before combining them back together.

## 5. How can I practice and improve my understanding of the chain rule?

One way to practice and improve your understanding of the chain rule is to work through various examples and exercises, either on your own or with the help of a tutor or study group. You can also use online resources, such as tutorials and interactive quizzes, to reinforce your understanding and identify any areas that may need more practice.

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