Problem 47 - Calculus Single Variable

Question

In each of Exercises \(45-50\) use the Chain Rule repeatedly to determine the derivative with respect to \(x\) of the given expression. $$ \cos ^{3}(2 x) $$

Step-by-Step Solution

Answer

The derivative is \(-6(\cos(2x))^2 \sin(2x)\).

Step 1: Identify the Outer and Inner Functions

The expression given is \( \cos^3(2x) \). Here, the outer function is \( u^3 \), where \( u = \cos(2x) \). The inner function is \( \cos(2x) \). Another inner function of \( \cos(2x) \) is \( 2x \).

Step 2: Differentiating the Outer Function

Firstly, find the derivative of the outer function with respect to \( u \). The derivative of \( u^3 \) is \( 3u^2 \).

Step 3: Differentiate the Middle Function

Next, differentiate the inner function \( \cos(2x) \) with respect to \( 2x \). The derivative of \( \cos(v) \) with respect to \( v \) is \( -\sin(v) \), hence the derivative of \( \cos(2x) \) is \( -\sin(2x) \).

Step 4: Differentiate the Innermost Function

Finally, differentiate the innermost function \( 2x \) with respect to \( x \). The derivative is \( 2 \).

Step 5: Apply the Chain Rule

Applying the chain rule, multiply all the above results. \( \frac{d}{dx}[\cos^3(2x)] = \frac{d}{du}[u^3] \times \frac{d}{dv}[\cos(2x)] \times \frac{d}{dx}[2x] = 3(\cos(2x))^2 (-\sin(2x)) \times 2 \).

Step 6: Simplify the Expression

Simplify the expression from Step 5: \(= -6(\cos(2x))^2 \sin(2x) \). This is the derivative of \( \cos^3(2x) \) with respect to \( x \).

Key Concepts

Understanding the Derivative

The concept of a derivative can be thought of as the mathematical equivalent of finding out how something changes. In simple terms, it's a measure of how a function's output value changes as its input value changes. Imagine when you are driving a car on a straight road; the speedometer shows how your position changes over time.
The derivative of a function indicates its rate of change:

• If a function is changing quickly, its derivative will have a larger value.
• If the function is not changing, its derivative will be zero.

In calculus, derivatives are used to understand curves and slopes. It's like knowing how steep a hill is while hiking.

What is an Outer Function?

In the realm of the chain rule, distinguishing between the outer and inner functions becomes essential. The outer function is essentially the function that appears on the outside of the given composition. In our initial exercise with the expression \( ext{cos}^3(2x)\), think of the outer function as the container or wrapper around the other functions; this is \(u^3\) as \(u = \text{cos}(2x)\).

• Outer functions are usually the last operations applied to whatever expression you're evaluating.
• In expressions like \((...)^n\), the power function tends to be the outer function.

The intent is to understand how changes in the inner function will affect the outer result.

Exploring the Inner Function

The inner function in a problem like this resides within another function. For our expression \( ext{cos}^3(2x)\), the inner function is \( ext{cos}(2x)\). An inner function determines what the outer function operates on.

• In terms of layers, the inner function resides deeper within the original formula.
• After the innermost functions, the next layer of our function tree involves the direct operation on the variable \(x\).

Consider the inner function as the heart of the operation. It's crucial to find its derivative to apply the chain rule effectively.

How Differentiation Works

Differentiation is the process of finding a derivative. Essentially, it involves calculating how the output of a particular function changes with respect to a particular input variable. Using the chain rule to differentiate composite functions helps make complex differentiation easier by breaking it down into manageable parts.
The process usually requires three steps:

• Identify and differentiate the outer function.
• Find the derivative of the inner function.
• Multiply the derivatives according to the chain rule.

Through differentiation, especially with the chain rule, we analyze how even slight changes in certain variables can lead to larger responses in our overall function, honing in on the intricacies of how functions behave.