Question

Compute the derivative of the given function. $$f(x)=e^{x} \ln x$$

Short answer

The derivative of the function is \( f'(x) = e^x \left(\frac{1}{x} + \ln x\right) \).

Step-by-step solution

Identify the function components

The given function is a product of two functions: \( u(x) = e^x \) and \( v(x) = \ln x \). To differentiate \( f(x) = e^x \ln x \), we'll use the product rule where \( (uv)' = u'v + uv' \).

Differentiate each component

First, differentiate \( u(x) = e^x \). The derivative \( u'(x) = e^x \).Next, differentiate \( v(x) = \ln x \). The derivative \( v'(x) = \frac{1}{x} \).

Apply the product rule

Using the product rule, substitute the derivatives:\[ (e^x \ln x)' = e^x \cdot \frac{1}{x} + e^x \ln x \cdot e^x \]This simplifies to:\[ \frac{e^x}{x} + e^x \ln x \]

Simplify the expression

Combining terms, the derivative \( f'(x) = e^x \left(\frac{1}{x} + \ln x\right) \) is the simplified form of the expression.

Key concepts

Understanding Derivatives

Derivatives are fundamental in calculus, representing the rate at which a function is changing at any given point. They form the basis for understanding motion, growth, and other dynamic changes in mathematics and applied sciences. When you take the derivative of a function, you are finding its slope, or its rate of change. This is often visualized as the tangent line at any point on the graph of the function.

Here’s why it’s important:

• It helps in finding where functions increase or decrease.
• It is used to find maximum or minimum points on graphs.
• It allows you to understand concavity and points of inflection on curves.

In the exercise, you were asked to compute the derivative of the function \( f(x) = e^x \ln x \). To tackle this, recognizing the function components is key. Differentiation requires understanding not only each component's individual derivative but also how they interact when multiplied together.

Applying the Product Rule

The product rule is a technique in calculus used to find the derivative of products of two functions. When one function is "multiplying" another, like \( e^x \) and \( \ln x \) in this exercise, the product rule states that:

• If \( u(x) \) and \( v(x) \) are two functions, their product \( uv \) has a derivative given by: \((uv)' = u'v + uv' \).

It’s a way to correctly account for the contribution of both functions to the rate of change.

In the solution, each function was treated separately:

• First, differentiate \( u(x) = e^x \). The derivative is also \( e^x \) because the exponential function is unique in this way.
• Second, differentiate \( v(x) = \ln x \). The derivative of \( \ln x \) is \( \frac{1}{x} \), a standard result in calculus.
• Finally, combine these derivatives using the product rule to get the overall derivative.

This ensures that each component is appropriately considered and combined.

Working with Natural Logarithms

The natural logarithm, denoted as \( \ln x \), is the logarithm to the base \( e \), where \( e \approx 2.71828 \). It is a fundamental function with natural occurrences in mathematics and science. In calculus, it is particularly significant:

• It arises when dealing with exponential growth and decay because of its relationship with the exponential function.
• Its properties make it very useful for simplifying complex expressions and integrals.

The derivative of \( \ln x \) is \( \frac{1}{x} \). This is essential when applying rules like the product rule, as it transforms \( \ln x \) into a form that can be combined with other derivatives. For our problem, \( \ln x \) works with \( e^x \) to produce both a product that requires the product rule and a simple form when differentiating.

Exploring the Exponential Function

The exponential function, written as \( e^x \), is known for its unique property that the derivative of \( e^x \) is itself, \( e^x \). This makes it incredibly efficient in calculus as differentiating it doesn’t change its form.

Consider its characteristics:

• Rapid growth: \( e^x \) grows faster than any polynomial as \( x \) increases.
• Application in modeling natural phenomena such as population growth, radioactive decay, and interest calculations.
• Simplicity in derivatives: The derivative of \( e^x \) maintains the function's simplicity.

In the exercise, \( e^x \) was one of the components being multiplied by \( \ln x \). Its straightforward derivative facilitates calculations and simplifies the process of working with the product rule. The exponential function is a beautiful example of the elegance of calculus, where patterns keep emerging seamlessly.