Problem 301 - Calculus Volume 1

Question

For the following exercises, find the derivative dy/dx. (You can use a calculator to plot the function and the derivative to confirm that it is correct.) [T] \(y=x \ln (x)\)

Step-by-Step Solution

Answer

The derivative of \( y = x \ln(x) \) is \( \frac{dy}{dx} = \ln(x) + 1 \).

Step 1: Identify the Function Type

The given function is \( y = x \ln(x) \). This is a product of two functions: \( f(x) = x \) and \( g(x) = \ln(x) \). We will use the product rule to find the derivative.

Step 2: Recall the Product Rule

The product rule states that for two functions \( f(x) \) and \( g(x) \), the derivative \( \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \).

Step 3: Differentiate Each Function

Find the derivative of each component:- \( f(x) = x \) implies \( f'(x) = 1 \).- \( g(x) = \ln(x) \) implies \( g'(x) = \frac{1}{x} \).

Step 4: Apply the Product Rule

Substitute into the product rule: \[ \frac{d}{dx}[x \ln(x)] = 1 \cdot \ln(x) + x \cdot \frac{1}{x} \].

Step 5: Simplify the Expression

Simplify the expression obtained in Step 4:- \( 1 \cdot \ln(x) = \ln(x) \)- \( x \cdot \frac{1}{x} = 1 \)Thus, \( \frac{d}{dx}[x \ln(x)] = \ln(x) + 1 \).

Key Concepts

Product Rule

The product rule is an essential technique in calculus for differentiating products of two functions. If you have a function that can be expressed as a product of two simpler functions, say \( f(x) \) and \( g(x) \), you can find its derivative efficiently using this rule. The product rule is mathematically stated as: \[\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\]This means that you take the derivative of the first function \( f(x) \), multiply it by the second function \( g(x) \), and then add this to the first function \( f(x) \) multiplied by the derivative of the second function \( g(x) \). This allows you to break down more complicated functions into smaller components that are easier to differentiate. This rule is crucial when dealing with exercises such as finding the derivative of \( y = x \ln(x) \), where the function is a product of \( x \) and \( \ln(x) \).

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which one quantity changes with respect to another, often thought of as the slope of the tangent line to the function at a given point. Differentiation is a fundamental concept in calculus that facilitates understanding of how functions change and behave.When performing differentiation, as in the exercise where \( y = x \ln(x) \), it's important to be familiar with basic differentiation rules such as the power rule and the chain rule, alongside the product rule. This allows you to find derivatives effectively and accurately. Understanding these rules, you can take the derivative of each term separately and then apply necessary operations to get the final derivative.

Natural Logarithm Derivative

The natural logarithm, denoted as \( \ln(x) \), is a significant mathematical function often encountered in calculus. The derivative of \( \ln(x) \) is one of the fundamental derivatives that you will use frequently. It is defined as:\[\frac{d}{dx} \ln(x) = \frac{1}{x}\]This simple derivative comes in handy when applying rules such as the product rule or chain rule. In the exercise \( y = x \ln(x) \), knowing that the derivative of \( \ln(x) \) is \( 1/x \) allows you to directly substitute this into the formula when you're calculating the derivative. Recognizing how to differentiate \( \ln(x) \) effectively is crucial for solving a wide range of problems in calculus.