Problem 1 - Single Variable Calculus

Question

Identify the "inside function" \(u=f(x)\) and the "outside function" \(y=g(u) .\) Then find \(d y / d x\) using the Chain Rule $$ y=(2 x+4)^{3} $$

Step-by-Step Solution

Answer

In the composite function \(y=(2x+4)^3\), the inside function is \(u=f(x)=2x+4\) and the outside function is \(y=g(u)=u^3\). To find the derivative \(\frac{dy}{dx}\) using the Chain Rule, we first find the derivatives of the inside and outside functions: \(\frac{du}{dx}=2\) and \(\frac{dy}{du}=3u^2\). Then, apply the Chain Rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = (3u^2) \cdot 2 = 6(2x+4)^2\).

Step 1: Identify the inside function (u=f(x)) and the outside function (y = g(u))

To identify the inside function and the outside function from the composite function \(y = (2x+4)^3\), we can see that the function inside the parentheses can be considered as the inside function, while the exponentiation operation outside the parentheses can be considered as the outside function. Therefore, we have the inside function as \(u = f(x) = 2x + 4\) and the outside function as \(y = g(u) = u^3\).

Step 2: Find the derivative of the inside function (u=f(x)) with respect to x

We need to find the derivative of the inside function (\(u = 2x + 4\)) with respect to x. By using the power rule, we can find the derivative: \[ \frac{du}{dx} = \frac{d}{dx}(2x + 4) = 2 \]

Step 3: Find the derivative of the outside function (y=g(u)) with respect to u

We need to find the derivative of the outside function (\(y = g(u) = u^3\)) with respect to u. Again, by using the power rule, we can find the derivative: \[ \frac{dy}{du} = \frac{d}{du}(u^3) = 3u^2 \]

Step 4: Apply the Chain Rule to find the derivative of y with respect to x

The Chain Rule states that if we have a composite function \(y = g(f(x))\), its derivative with respect to x can be found as: \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \] Now, substituting the values of the derivatives \(\frac{dy}{du}\) and \(\frac{du}{dx}\) from Steps 2 and 3, and also the expression for the inside function \(u = 2x + 4\), we can find \(\frac{dy}{dx}\) as follows: \[ \frac{dy}{dx} = (3u^2) \cdot 2 = 3(2x+4)^2 \cdot 2 \] So the final answer is, \[ \frac{dy}{dx} = 6(2x+4)^2 \]

Key Concepts

Inside Function

When dealing with a composite function like \( y = (2x+4)^3 \), understanding the inside function is crucial. The inside function is the part that is first addressed in the calculation—usually located within parentheses or deeply nested.

**In practical terms**:

• Look for expressions inside parentheses or brackets within a function.
• It is often the piece of the function where the variable \( x \) appears directly.

In our example, the expression \( 2x+4 \) is the inside function, noted as \( u = f(x) = 2x+4 \). This needs to be solved prior to considering the outer layers of the function.

Knowing the inside function is essential for taking derivatives using the Chain Rule. This understanding comes from recognizing it as the component directly influenced by \( x \), and its derivative will be calculated with respect to \( x \).

Outside Function

Let's turn our attention to the outside function, which encompasses the overall operation that affects the inside function. It is applied to the value of the inside function.

In our example, the entire expression \( (2x+4)^3 \) shows how the inside function \( 2x+4 \) is associated with an exponent. Hence, the outside function is \( y = g(u) = u^3 \).

**Identifying the outside function**:

• It is usually applied after the inside function.
• This component can include exponents, trigonometric functions, or any other operation acting on the result of the inside function.

Understanding the arrangement of the outside function lets us comprehend how the composite function behaves entirely. This forms an essential part of deriving the composite function by breaking it down part by part.

Composite Function

Now that we have the inside and outside functions, it's time to see how they come together as a composite function. A composite function integrates these two elements, combining them to form a single, unified function. In our case, the composite function looks like this:\[ y = (2x+4)^3\]

**Thinking in terms of composites**:

• The inside function \( f(x) \) is evaluated first.
• Then, the result of \( f(x) \) is plugged into the outside function \( g(u) \), replacing \( u \) with \( f(x) \).

They work in tandem: the function within guides the initial value, and then the external operation is applied to the result. By mastering identifying this structure, the Chain Rule's applications will become much clearer. This makes complex derivative calculations far simpler.

Derivative Calculation

Having broken down the functions, let's move to find the derivative using the powerful Chain Rule. The Chain Rule lets us find the derivative of composite functions by taking derivatives step by step.

**Steps to find the derivative**:

• First, differentiate the inside function: \( \frac{du}{dx} = 2 \).
• Next, differentiate the outside function with respect to \( u \): \( \frac{dy}{du} = 3u^2 \).
• Apply the Chain Rule: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

Fill in the derivatives:\[ \frac{dy}{dx} = (3u^2) \cdot 2 = 3(2x+4)^2 \cdot 2\] So, finally, the derivative of our composite function is\[ \frac{dy}{dx} = 6(2x+4)^2\]Understanding these steps ensures that you can tackle derivatives of more complicated functions confidently.