Problem 46 - Calculus and its applications

Question

Differentiate each function. \(f(x)=\frac{3 x^{2}+2 x}{x^{2}+1}\)

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{2x^2 + 6x + 2}{(x^2 + 1)^2} \).

Step 1: Differentiate Using the Quotient Rule

To differentiate the function \(f(x) = \frac{3x^2 + 2x}{x^2 + 1}\), use the quotient rule, which states: if \(y = \frac{u}{v}\), then \(y' = \frac{u'v - uv'}{v^2}\). In this case, \(u = 3x^2 + 2x\) and \(v = x^2 + 1\).

Step 2: Differentiate the Numerator

Find the derivative of the numerator \(u = 3x^2 + 2x\). This is done by applying the power rule: \(u' = 6x + 2\).

Step 3: Differentiate the Denominator

Find the derivative of the denominator \(v = x^2 + 1\). This is also done using the power rule: \(v' = 2x\).

Step 4: Apply the Quotient Rule Formula

Substitute the derivatives found in Steps 2 and 3 into the quotient rule formula: \[ f'(x) = \frac{(6x + 2)(x^2 + 1) - (3x^2 + 2x)(2x)}{(x^2 + 1)^2} \]

Step 5: Simplify the Resulting Expression

Expand and simplify the expression: \( f'(x) = \frac{(6x^3 + 6x + 2x^2 + 2) - (6x^3 + 4x^2)}{(x^2 + 1)^2} \) Combine like terms: \( f'(x) = \frac{2x^2 + 6x + 2}{(x^2 + 1)^2} \)

Step 6: Final Form of the Derivative

The simplified derivative is given by: \[ f'(x) = \frac{2x^2 + 6x + 2}{(x^2 + 1)^2} \].

Key Concepts

Differentiation

Differentiation is a crucial concept in calculus, representing the process of finding a derivative. In simple terms, a derivative measures how a function changes as its input changes. This is particularly useful for understanding rates of change and slopes of curves at any given point.When you differentiate a function, you apply specific rules and techniques to determine its rate of change. For the function given, \[ f(x)=\frac{3x^2+2x}{x^2+1} \]differentiation involves using several rules, making it an interesting example. Key steps in differentiation include identifying each part of the function and appropriately applying rules. Here, the Quotient Rule is used due to the function's fraction form. Overall, mastering differentiation equips you with a powerful tool to explore various mathematical and real-world problems.

Power Rule

The Power Rule is one of the foundational tools in differentiation. It simplifies the process of finding the derivative of a polynomial. The rule states that if you have a term \(x^n\), its derivative is \(nx^{n-1}\). For example, in the numerator of our function, \(3x^2 + 2x\), the Power Rule is applied as follows:

• The derivative of \(3x^2\) is \(6x\).
• The derivative of \(2x\) is \(2\).

This gives the derivative of the numerator as \(6x + 2\).Similarly, for the denominator, \(x^2 + 1\), applying the Power Rule yields:

• The derivative of \(x^2\) is \(2x\).
• The constant \(1\) has a derivative of \(0\), as constants do not change.

This gives the derivative of the denominator as \(2x\).Mastering the Power Rule streamlines the initial steps in solving more complex calculus problems that involve polynomials.

Numerical Simplification

Numerical simplification is the process of cleaning up mathematical expressions to make them as simple as possible. After applying the quotient and power rules, your solution will often need further refinement to express the derivative cleanly and accurately.For the function:\[ f'(x) = \frac{(6x + 2)(x^2 + 1) - (3x^2 + 2x)(2x)}{(x^2 + 1)^2} \]simplification involves expanding the products and then combining like terms.Going through the steps:

• Expand both terms in the numerator: you distribute all components with their partners in the expressions.
• Combine like terms: gather and add or subtract terms of the same degree.

This step ensures the expression is concise, reducing potential errors in further calculations or interpretations.An essential skill in mathematics is simplifying expressions efficiently, highlighting numerical simplification as a critical phase in calculus, especially when dealing with complex derivatives.