Problem 31 - Calculus Single Variable

Question

Use the Quotient Rule to compute the derivative of the given expression with respect to \(x .\) $$ (5 x-2 / x) / x $$

Step-by-Step Solution

Answer

Derivative: \( \frac{4}{x^3} \)

Step 1: Simplify the Expression

The original expression is \( \frac{5x - \frac{2}{x}}{x} \). Start by rewriting it as \( \frac{5x^2 - 2}{x^2} \) to make the application of the Quotient Rule clearer, where the numerator is \( 5x^2 - 2 \) and the denominator is \( x^2 \).

Step 2: Identify the Functions

Let \( u(x) = 5x^2 - 2 \) and \( v(x) = x^2 \). These are the functions in the form required by the Quotient Rule, which is for expressions of the form \( \frac{u(x)}{v(x)} \).

Step 3: Compute the Derivatives

Take the derivative of each identified function: \( u'(x) = \frac{d}{dx}(5x^2 - 2) = 10x \) and \( v'(x) = \frac{d}{dx}(x^2) = 2x \).

Step 4: Apply the Quotient Rule

The Quotient Rule states that the derivative of \( \frac{u}{v} \) is \( \frac{u'v - uv'}{v^2} \). Substitute the derivatives: \( \frac{10x \cdot x^2 - (5x^2 - 2) \cdot 2x}{(x^2)^2} \).

Step 5: Simplify the Expression

Perform the multiplications in the numerator: \( 10x^3 - (10x^3 - 4x) \). Substituting gives \( \frac{10x^3 - 10x^3 + 4x}{x^4} \). This simplifies to \( \frac{4x}{x^4} = \frac{4}{x^3} \).

Key Concepts

Derivative Calculation

Calculating derivatives is a fundamental part of calculus. Derivatives help us find the rate at which a function is changing at any given point. To calculate derivatives, you need to understand the different rules of differentiation.
One commonly used rule is the Quotient Rule, which is especially useful for expressions that are fractions, such as \( \frac{u(x)}{v(x)} \). In this exercise, we apply the Quotient Rule to solve the given problem.
The general formula for the Quotient Rule is:

• \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \)

Here, \( u' \) and \( v' \) represent the derivatives of \( u(x) \) and \( v(x) \) respectively. This formula helps us find the derivative of a function that is expressed as the ratio of two other functions.

Simplifying Expressions

Simplifying expressions is a crucial step in solving calculus problems as it makes the process of applying rules much easier. In this problem, we began with the expression \( \frac{5x - \frac{2}{x}}{x} \). It initially might seem complicated, but simplifying it can illuminate the path forward.
The key is to rewrite complicated parts into simpler forms. Here, by multiplying terms appropriately, the expression is restructured as \( \frac{5x^2 - 2}{x^2} \). This step is necessary before applying the Quotient Rule because it requires a clear defined "top" \( u \) and "bottom" \( v \) for the expression.

• Rewriting expressions often involves basic algebraic manipulations such as multiplying through by a common denominator or expanding brackets.

Simplifying helps in managing and visualizing the expression better, thus enabling smoother solving.

Function Derivatives

The concept of function derivatives involves finding how a function changes as its variable changes. In simpler terms, it gives us the slope of the function at any given point, which is its rate of change.
When applying the Quotient Rule, computing the derivatives of the functions involved is a vital step. Given \( u(x) = 5x^2 - 2 \) and \( v(x) = x^2 \), their derivatives are found as follows:

• \( u'(x) = \frac{d}{dx}(5x^2 - 2) = 10x \)
• \( v'(x) = \frac{d}{dx}(x^2) = 2x \)

Finding these derivatives accurately is critical because they directly influence the outcome when substituted back into the Quotient Rule.
This stage of the process lays the groundwork for computing the full derivative of the original function.

Fractional Expressions

Fractional expressions often pose specific challenges and opportunities when calculating derivatives. When dealing with something like \( \frac{5x^2 - 2}{x^2} \), using the Quotient Rule is essential.
The numerator and denominator of the expression, \( 5x^2 - 2 \) and \( x^2 \) respectively, require careful handling. By applying derivatives to both, the simplified form can be manipulated for further calculation:

• The derivative of the numerator, \( u'(x) = 10x \), influences the overall expression significantly.
• Similarly, the derivative of the denominator, \( v'(x) = 2x \), is crucial for applying the Quotient Rule.

Once you substitute these into the Quotient Rule, the result is simplified from a potentially complex form to a more manageable one, ultimately yielding the derivative as \( \frac{4}{x^3} \). This process highlights how fractional expressions, though initially tricky, can be deftly handled to reveal their rates of change.