The Chain Rule Activating Strategy

​How is Lindt Chocolate Made?

The process of making chocolate is quite involved and includes multiple steps. In large chocolate factories, this process has many automated components and uses an assembly line. At each station, a single component of the process is completed before it is sent off to the next station in the chain.

Reorder

The steps for making chocolate bars are jumbled below.

Put them in the correct order by numbering each step from 1 to 4.

• Roasting and crushing the beans

• Mixing in other dry ingredients (milk powder, vanilla, sugar)

• Conching and tempering (heating and smoothing out the chocolate paste)

• Packaging

1

2

3

4

• 1
• 2
• 3
• 4

Labeling

List the inputs and outputs at each step in the chocolate making process.

Drag labels to their correct position on the image

Crushed Beans

Paste

Beans

Chocolate

Lindt Package

​Notice that the output of one step becomes the input of the next step

​Here is the solution to the previous slide

Math Response

We are going to use the assembly line approach to analyze a complex function with multiple parts. Let’s look at the function $$y=\sin\left(x^2-3\right)$$

The raw material is our input, or independent variable. What is the independent variable of this function?

Type answer here

Correct Answer: x

Deg^°

Rad

Reorder

Imagine evaluating $$y=\sin\left(x^2-3\right)$$ at x=5.

What happens first in the assembly line?

What happens next?

Put the steps below in the correct order.

• Substitute 5 in for x.

• Square it

• Subtract 3 from it

• Take the sine of it

1

2

3

4

• 1
• 2
• 3
• 4

Math Response

To find the derivative of this function, we will need to consider how each link in the chain contributes to the end product.

If $$\nabla=x^2-3$$ , what is $$\frac{d\nabla}{dx}?$$

Type answer here

Correct Answer: 2x

Deg^°

Rad

Fill in the Blanks

Plugging $\nabla=x^2-3$ back into our original function, we see that $y=\sin\nabla$ .

What is $\frac{dy}{d\nabla}?$

Fill in the Blanks

Now to get our final product we multiply the two answers from the previous steps.

So $\frac{d\nabla}{dx}\cdot\frac{dy}{d\nabla}$ .

What does this product simplify to if we treat it like multiplying fractions?

​Here are the main takeaways from this activity:

Multiple Choice

Your turn:

Differentiate using the Chain Rule.

Then select the correct answer.

• -2sin(2x)

• -sin(2x)

• sin(2x)

• 2sin(2x)

• 2sin(x)

Math Response

If $$y=\left(x^2-4\right)^5$$

Then $$\frac{dy}{dx}=$$

Type answer here

Correct Answer: 5\left(x^2-4\right)^4\left(2x\right)

Deg^°

Rad

​Great job!!!
You finished the Chain Rule Activating Strategy