Problem 8 - Calculus - AP Edition

Question

Find \(d y / d x\) $$y=\frac{x^{2}+1}{5}$$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{2x}{5} \).

Step 1: Identify the Function

Given: \( y = \frac{x^{2} + 1}{5} \). This is a simple fraction where the numerator is a polynomial and the denominator is a constant.

Step 2: Simplify the Expression

Rewrite the function as a multiplication with a constant factor: \( y = \frac{1}{5}(x^{2} + 1) \).

Step 3: Apply the Derivative Rule for Constants

Start differentiating using the constant rule. Since \( \frac{1}{5} \) is a constant, it remains as it is: \( \frac{d}{dx}(\frac{1}{5} g(x)) = \frac{1}{5} \frac{d}{dx}(g(x)) \), where \( g(x) = x^2 + 1 \).

Step 4: Differentiate the Inner Function

Differentiate \( g(x) = x^2 + 1 \) with respect to \( x \). The derivative of \( x^2 \) is \( 2x \), and the derivative of \( 1 \) is 0. So, \( \frac{d}{dx}(x^2+1) = 2x \).

Step 5: Apply the Derivative Result

Substitute the derivative of the inner function back: \( \frac{d}{dx}(y) = \frac{1}{5} \cdot 2x = \frac{2x}{5} \).

Step 6: Final Answer

The derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = \frac{2x}{5} \).

Key Concepts

Derivative of a Polynomial

When we talk about finding the derivative of a polynomial, we're looking at how to determine the rate at which a polynomial function changes. Polynomials are functions like \( x^2 + 1 \), which appear frequently in math. Calculating the derivative of a polynomial involves applying derivative rules to each term separately, identifying how each individual component of the polynomial contributes to the overall change.

To differentiate a polynomial, apply the power rule to each term. For example, the power rule tells us that

• the derivative of \( x^n \) is \( nx^{n-1} \).

In our exercise, we have \( x^2 + 1 \). Applying the power rule gives us:

• The derivative of \( x^2 \) is \( 2x \)
• The derivative of a constant like \( 1 \) is \( 0 \).

Thus, the derivative of the polynomial \( x^2 + 1 \) is simply \( 2x \). This step is crucial in understanding how functions behave and change.

Constant Factor in Differentiation

Differentiating expressions with a constant factor involves a straightforward rule which makes it easier to calculate derivatives. When you have a constant multiplied by a function, this constant can be "taken out" of the differentiation process.

If you have a function \( y = c \, f(x) \), where \( c \) is a constant, the derivative will be \( c \cdot f'(x) \).
This means you first find the derivative of the function \( f(x) \), then multiply this derivative by the constant.

In our example, the function \( y = \frac{1}{5}(x^2 + 1) \) has a constant factor \( \frac{1}{5} \). To differentiate this expression, we can apply the constant factor rule:

• Differentiate \( x^2 + 1 \) to get \( 2x \).
• Multiply it by the constant \( \frac{1}{5} \).

Hence, the derivative is \( \frac{1}{5} \cdot 2x = \frac{2x}{5} \). This approach streamlines the differentiation process, making it efficient and systematic.

Basic Differentiation Rules

Basic differentiation rules are fundamental tools that help us easily figure out the derivatives of common functions.

These rules provide a framework for systematically finding derivatives without needing to compute limits every time.
Here are a few key rules:

• The Power Rule: For any function \( x^n \), its derivative is \( nx^{n-1} \).
• The Constant Rule: The derivative of a constant is zero.
• Constant Multiple Rule: If a function \( f(x) \) is multiplied by a constant \( c \), its derivative is \( c \cdot f'(x) \).

Using these rules simplifies the process of finding derivatives for complex functions by breaking them down into simpler components.
This makes it easier for any student to approach differentiation in a step-by-step manner, as seen in our problem where these rules helped transform \( y = \frac{x^2 + 1}{5} \) into an easy task.