91Ó°ÊÓ

When tackling calculus problems involving derivatives of products of functions, the product rule is essential. For any two differentiable functions, u(x) and v(x), the product rule states that the derivative of their product u(x) * v(x) with respect to x is given by:

\[ \frac{d}{dx}[u(x) * v(x)] = u'(x) * v(x) + u(x) * v'(x) \]

In simpler terms, you take the derivative of the first function and multiply it by the second function 'as is', and then add the product of the first function 'as is' and the derivative of the second function. For the exercise at hand, where the function is y = x^2 \ln x, u(x) is x^2, and v(x) is \ln x. Using the product rule, one part of the derivative comes from differentiating x^2 (which is 2x) and leaving \ln x alone, and the other part comes from keeping x^2 and differentiating \ln x (which gives 1/x). Added together, they provide the complete derivative of the product, illustrating the product rule in action.

Derivatives are a fundamental concept in differential calculus, representing how a function changes as its input changes. Simply put, the derivative of a function at a certain point quantifies the instant rate of change of the function with respect to its variable at that point. It's like looking at the speedometer of a car at a particular instant.

For a function y = f(x), the derivative with respect to x is denoted as f'(x) or \(\frac{dy}{dx}\). It can be visually interpreted as the slope of the tangent line to the graph of the function at that point. In our exercise, the derivative of y = x^2 \ln x at x=1 is computed using the product rule, and it serves to determine how y changes when x is perturbed slightly from 1 to 1.01, an essential calculation in many fields of science and engineering that deal with small changes over time or space.

Logarithmic differentiation is a technique used when differentiating complex functions, especially those involving products, quotients, or powers that are difficult to handle using standard rules. In this technique, you take the natural logarithm (ln) of both sides of an equation y=f(x) and then differentiate with respect to x. It simplifies the differentiation process because the properties of logarithms convert multiplication into addition, division into subtraction, and powers into multiplication.

To apply this method, you would proceed as follows:

• Take the natural logarithm of both sides: ln(y) = ln(f(x)).
• Use the properties of logarithms to simplify the right side.
• Differentiate both sides with respect to x, remembering to use the chain rule on the left side (d/dx[ln(y)] = 1/y * dy/dx).
• Finally, you'll solve for dy/dx to find the derivative.

In the context of the example given in the exercise, logarithmic differentiation would make it easier to handle the derivative because it could simplify the expression before applying the product rule. It does not, however, change the derivative's value—it's just another method to find the same outcome.