Question

$$ \text { } , \text { find the indicated derivative. } $$ $$ \frac{d y}{d x} \text { if } y=x^{2} \ln x $$

Short answer

The derivative is \( y' = 2x \ln x + x \).

Step-by-step solution

Identify the types of functions involved

The function given is a combination of a polynomial function, \(x^2\), and a logarithmic function, \(\ln x\). The product rule for derivatives is applicable here because we have a product of two functions.

Recall the Product Rule

The product rule is used to find the derivative of a product of two functions. If \(u(x)\) and \(v(x)\) are two functions, then the derivative of their product is given by: \(\frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v'\).

Assign functions for differentiation

Let \(u(x) = x^2\) and \(v(x) = \ln x \). Then, according to the product rule, we need to find the derivatives \(u'(x)\) and \(v'(x)\).

Differentiate \(u(x) = x^2\)

Use the power rule: \(u'(x) = \frac{d}{dx}(x^2) = 2x\).

Differentiate \(v(x) = \ln x\)

Use the basic derivative rule for the natural logarithm: \(v'(x) = \frac{d}{dx}(\ln x) = \frac{1}{x}\).

Apply the Product Rule and Substitute

Using the product rule, calculate \(y' = u' \cdot v + u \cdot v'\). Substituting the derivatives, we have: \(y' = (2x)(\ln x) + (x^2)(\frac{1}{x})\).

Simplify the expression

Calculate each term: \((2x)(\ln x) = 2x \ln x\) and \((x^2)(\frac{1}{x}) = x\). Therefore, the derivative is: \(y' = 2x \ln x + x\).

Key concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus, capturing the idea of how a function changes as its input changes. Imagine you're observing how the position of a car on a track changes over time; the derivative gives you the car's speed at any moment. It essentially measures the rate at which something happens, which can be very useful in a variety of fields like physics, economics, and engineering.

When working with derivatives, we can deal with simple functions like powers of x or more complex combinations involving products or quotients. For a function defined as a product of two other functions, the product rule is our go-to tool. It helps us find the derivative of their product by considering the rate of change of each function separately, then combining these rates in a specific way.

By using derivatives, we can gain insights into the slope of a function at any point, indicating whether our function is increasing or decreasing. This makes it possible to identify peaks, troughs, or any point where the function changes behavior.

Essentials of Polynomial Functions

Polynomial functions are one of the simplest and most common types of functions, composed of terms that are non-negative integer powers of the variable, usually represented as x. For example, a simple polynomial function might look like: \(f(x) = x^2 + 3x + 5\). These functions are easy to manage in terms of differentiation due to the power rule.

• The power rule states that for any term \(x^n\), its derivative is \(nx^{n-1}\).

In our exercise, we apply this rule to the function \(x^2\), giving us a derivative of \(2x\). This tells us how fast the \(x^2\) component is changing for small changes in x.

Polynomial functions are smooth and continuous, making them extremely predictable and useful for modeling data, fitting curves, and predicting outcomes within a safe domain.

The Role of Logarithmic Functions

Logarithmic functions are another essential set of functions encountered in calculus, known for their unique properties and their inverse relationship to exponential functions. The natural logarithm, denoted as \(\ln x\), is particularly important because it is the inverse of the exponential function \(e^x\).

In calculus, the derivative of a logarithmic function is both simple and elegant: the derivative of \(\ln x\) is \(\frac{1}{x}\). This result stems from its unique properties and helps solve many real-world problems involving growth rates and compounded changes.

• Logarithmic functions appear in various fields such as science, where they help analyze phenomena that involve exponential growth or decay.

In our exercise, we use the product rule to differentiate a function that combines both a polynomial (\(x^2\)) and a logarithmic function (\(\ln x\)). By understanding how these two types of functions behave and change, we gain a more nuanced view of the entire system described by their product.