Problem 8 - Calculus Early Transcendentals

Question

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\left(5 x^{2}+11 x\right)^{20}$$

Step-by-Step Solution

Answer

Based on the given function $$y = (5x^2 + 11x)^{20}$$, the derivative $$\frac{dy}{dx}$$ can be calculated using the Chain Rule as follows: 1. Identify the inside and outside functions: $$u = 5x^2 + 11x$$ and $$y = u^{20}$$ 2. Find the derivative of the inside function: $$\frac{d}{dx}(5x^2+11x) = 10x + 11$$ 3. Find the derivative of the outside function: $$\frac{dy}{du} = \frac{d}{du}(u^{20}) = 20u^{19}$$ 4. Apply the Chain Rule: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = (20u^{19})\cdot(10x+11)$$ 5. Replace $$u$$ with the inside function: $$\frac{dy}{dx} = 20(5x^2+11x)^{19}(10x+11)$$

Step 1: Identify the inside and outside functions

Here, the inside function is $$u = 5x^2 + 11x$$ and the outside function is $$y = u^{20}$$.

Step 2: Find the derivative of the inside function

We need to find the derivative of $$u$$ with respect to $$x$$: $$\frac{d}{dx}(5x^2+11x) = 10x + 11$$

Step 3: Find the derivative of the outside function

Now, we need to find the derivative of $$y$$ with respect to $$u$$: $$\frac{dy}{du} = \frac{d}{du}(u^{20}) = 20u^{19}$$

Step 4: Apply the Chain Rule

Now, we will apply the Chain Rule by multiplying the derivative of the outside function with the derivative of the inside function: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = (20u^{19})\cdot(10x+11)$$

Step 5: Replace $$u$$ with the inside function

Substitute $$u$$ back into the expression with the original inside function: $$\frac{dy}{dx} = 20(5x^2+11x)^{19}(10x+11)$$

Key Concepts

Derivative

A derivative represents the rate at which a function changes. Think of it as the speedometer in your car showing how fast you're going at any moment. In calculus, when we talk about derivatives, we're interested in the slope of a function at any given point. This tells us how quickly the function is increasing or decreasing at that point.

To find the derivative, we use various rules, and the Chain Rule is one of them. In our exercise, we deal with a composite function which means there is a function within a function. The Chain Rule helps us find the derivative of such functions by breaking down the task into simpler parts.

• It involves two main functions: the inside function and the outside function.
• By differentiating these functions separately, we then multiply their results to find the final derivative.
• The final result tells us how the whole composite function is changing with respect to the original variable.

Inside Function

When dealing with composite functions and the Chain Rule, identifying the inside function is a crucial first step. The inside function is what is "nested" inside another function. For our problem:

• The inside function is given as \( u = 5x^2 + 11x \).
• It's like the core you need to unravel before you can deal with the entire expression.

Our task is to find the derivative of this inside function with respect to \(x\). Once identified, we use standard derivative rules. For \( u \, (5x^2 + 11x) \):
\[ \frac{d}{dx}(5x^2 + 11x) = 10x + 11\]
This derivative tells us the rate of change of the inside function as \(x\) varies.

Outside Function

The outside function in the context of the Chain Rule is the function in which the inside function is embedded. Think of it like a wrapping around the core function.

In our example, the outside function is based on the new variable \( u \) and is expressed as \( y = u^{20} \).

• To find its derivative, we differentiate with respect to \( u \) as though \( u \) is our main variable.
• Using the power rule, the derivative is calculated as \( \frac{dy}{du} = 20u^{19} \).

This step captures the potential growth due to the outer shell of the function structure and is multiplied with the inside function's derivative to get the complete rate of change.

Calculus

Calculus is a branch of mathematics focused on the study of change. It's split into two main ideas: differentiation (as explored in our example) and integration. In the realm of differentiation, the Chain Rule is an essential technique that enables us to find derivatives of composite functions.

In our exercise, the Chain Rule allows us to understand how the entire expression \( \left(5x^{2} + 11x\right)^{20} \) changes with \(x\). Here's how this critical process works:

• We identify both the inside and outside functions.
• Next, we compute their derivatives separately.
• Finally, we combine them through multiplication.

Calculus, and specifically the Chain Rule, provides a structured approach to navigate and solve problems involving these complex functions, revealing how intricate expressions depend on changes in their variables.