91Ó°ÊÓ

Differentiation is a fundamental tool in calculus that allows us to study how a function changes as its input changes. There are several key rules that help us find derivatives efficiently, and these are known as differentiation rules. Some basic ones include the power rule, product rule, and chain rule.

In our exercise, we're using some of these rules to differentiate the function \( y = \frac{1}{x} - \ln x \). To find the first derivative, we apply the following rules:

• The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \) by the power rule.
• The derivative of \( \ln x \) is \( \frac{1}{x} \), derived from the properties of logarithms.

Combining these results, we have the first derivative of \( y \): \( y' = -\frac{1}{x^2} - \frac{1}{x} \). Understanding these foundational rules sets the stage for more complex differentiation processes, such as higher order derivatives.

When taking derivatives of expressions involving division, we use the quotient rule. This rule is vital in our exercise when we find the second derivative of the function. The quotient rule states:

\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v u' - u v'}{v^2} \]

where \( u \) and \( v \) are differentiable functions of \( x \). In this case, when we calculate \( \frac{d^2 y}{d x^2} \), we set \( u = -(1+x) \) and \( v = x^2 \).

• The derivative of \( u \), \( u' \), is \(-1\).
• The derivative of \( v \), \( v' \), is \( 2x \).

Plugging these into the quotient rule formula helps us find the second derivative, which is then simplified in subsequent steps. Mastery of the quotient rule is crucial for tackling complex algebraic fractions and ensuring precision in higher-order differentiation.

Simplification is an often overlooked but essential part of working with derivatives. To make algebraic expressions more manageable and the calculations clearer, we simplify the derivatives whenever possible.

In our exercise, once we found the first derivative \( y' = -\frac{1}{x^2} - \frac{1}{x} \), we simplified it to \( -\frac{1 + x}{x^2} \). This step sets us up for a cleaner application of the quotient rule.

After calculating the second derivative while using the quotient rule, we simplify further as follows:

• The expression is \( \frac{-x^2 + 2x^2 + 2x^3}{x^4} \).
• Combine like terms to get \( \frac{x^2 + 2x^3}{x^4} \).

This final simplification makes the derivative expression neater and more comprehensible. Simplifying derivatives not only reduces computational work but also enhances understanding by revealing the essential parts of the function.