Problem 17 - Single Variable Calculus

Question

Find the derivative of each function. \(f(x)=\frac{2 \sqrt{x}}{x^{2}+1}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=\frac{2 \sqrt{x}}{x^{2}+1}\) is \(f'(x) =\frac{x^{\frac{1}{2}}(x^2 + 1) - 4x^{\frac{3}{2}}}{(x^2 + 1)^2}\).

Step 1: Identify the Quotient Rule

The Quotient Rule states the following: If \(f(x) = \frac{g(x)}{h(x)}\), then \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\). In our case, we have \(g(x) = 2\sqrt{x}\) and \(h(x) = x^2 + 1\). We need to find the derivatives of \(g(x)\) and \(h(x)\) to apply the Quotient Rule.

Step 2: Find g'(x) using Chain Rule

The Chain Rule states: \(f'(x) = \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)\). Rewrite \(g(x) = 2\sqrt{x}\) as \(g(x) = 2x^{\frac{1}{2}}\). Now, we apply the Chain Rule: \(g'(x) = \frac{d}{dx}(2x^{\frac{1}{2}}) = \frac{1}{2} \cdot 2x^{-\frac{1}{2}} = x^{-\frac{1}{2}}\).

Step 3: Find h'(x) using the Power Rule

The Power Rule states that if \(h(x) = x^n\), then \(h'(x) = nx^{n-1}\). \(h(x) = x^2 + 1\), so \(h'(x) = 2x^1 + 0 = 2x\).

Step 4: Apply the Quotient Rule

Now that we have found \(g'(x)\) and \(h'(x)\), we can apply the Quotient Rule: \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} =\frac{x^{-\frac{1}{2}}(x^2 + 1) - 2x^{\frac{1}{2}}(2x)}{(x^2 + 1)^2}\).

Step 5: Simplify the result

Now, we need to simplify the expression we found: \(f'(x) = \frac{x^{\frac{1}{2}}(x^2 + 1) - 4x^{\frac{3}{2}}}{(x^2 + 1)^2}\). Thus, the derivative of the given function is \(f'(x) =\frac{x^{\frac{1}{2}}(x^2 + 1) - 4x^{\frac{3}{2}}}{(x^2 + 1)^2}\).

Key Concepts

Chain Rule

The Chain Rule is a fundamental tool in calculus used to differentiate composite functions. Imagine one function nested within another, like a box within a box. The Chain Rule helps us understand how these functions relate.
The rule states that to differentiate a composite function, you differentiate the outer function and multiply it by the derivative of the inner function.

• Start by identifying the "outer" and "inner" functions in your expression.
• Differentiating step-by-step, tackle the outer function first.
• Compute the derivative of the inner function.
• Multiply them together for the final result.

In our exercise, the expression inside the square root is treated with the Chain Rule, simplifying the calculation of derivatives.

Power Rule

The Power Rule is a quick way to find the derivative of expressions raised to a power. If you have a function like \(x^n\), the Power Rule tells you its derivative is \(nx^{n-1}\).
Let's break it down:

• When the exponent \(n\) is positive, each differentiation lowers the power by one.
• Multiply by the original power to get the correct derivative.

In our problem, we encountered both positive and rational exponents.
Remember that a root, like in \(\sqrt{x}\), can be rewritten using fractional exponents (e.g., \(x^{1/2}\)). This helps apply the Power Rule effectively, as demonstrated in step-by-step solutions.

Derivative

A derivative represents the rate at which a function is changing at any given point. It plays a critical role in analyzing functions and understanding their behavior.
Basics of derivatives include:

• The slope of a function at any point.
• The speed of change as values shift.
• Mathematically, it's the limit of the difference quotient as the interval approaches zero.

In the exercise, finding the derivative tells us how quickly \(f(x)\) changes concerning \(x\).
Using the Quotient Rule along with the Chain and Power Rules, we calculated the derivative of the given fractional function.

Simplifying Expressions

Simplifying expressions is a key step in expressing derivatives neatly and understandably.
Post-calculation, it's essential to combine like terms and refine complex fractions. It ensures that:

• Results are easy to interpret and use.
• It leads to simplification checks which help verify correctness.

In our scenario, the derivative expression originally appeared complex, but by gathering similar terms and rearranging, we achieved a more elegant representation:
\(f'(x) =\frac{x^{1/2}(x^2 + 1) - 4x^{3/2}}{(x^2 + 1)^2}\)
Always aim for simplicity in your final derivatives!