Problem 9 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=x e^{x}\)

Step-by-Step Solution

Answer

The derivative of \(f(x) = x e^{x}\) with respect to \(x\) is \(f'(x) = e^{x}(1 + x)\).

Step 1: Identify the Rule

To differentiate the function \(f(x) = x e^{x}\), we should identify that this is a product of two functions: \(x\) and \(e^{x}\). We will use the product rule for differentiation.

Step 2: State the Product Rule

The product rule states that if you have a function \(u(x) v(x)\), then its derivative \(\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)\). For our function, \(u(x) = x\) and \(v(x) = e^{x}\).

Step 3: Differentiate First Component

Find the derivative of \(u(x) = x\). The derivative \(u'(x)\) is 1, since the derivative of \(x\) with respect to \(x\) is 1.

Step 4: Differentiate Second Component

Find the derivative of \(v(x) = e^{x}\). The derivative \(v'(x)\) is \(e^{x}\), since the derivative of \(e^{x}\) with respect to \(x\) is \(e^{x}\).

Step 5: Apply the Product Rule

Using the derivatives calculated, apply the product rule: \[ f'(x) = u'(x)v(x) + u(x)v'(x) = 1 \cdot e^{x} + x \cdot e^{x}. \]

Step 6: Simplify the Expression

Simplify the expression obtained in Step 5: \[ f'(x) = e^{x} + x e^{x} = e^{x}(1 + x). \] This is the derivative of \(f(x) = x e^{x}\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, used to measure how a function changes as its input changes. It is, essentially, the process of finding the derivative of a function. Think of differentiation as a tool to find how quickly or slowly a function is changing at any given point. For a function like a curve on a graph, the derivative at a point tells you the slope of the tangent line at that point.
Some key points about differentiation:

• The derivative represents the rate of change or the slope of the function.
• The process involves finding the limit of the difference quotient as the interval approaches zero.
• Derivatives can be calculated for both simple functions like polynomials and more complex expressions.

In the case of the exercise, the function we differentiated was a product of two simpler functions. This is a common scenario where the product rule, a specific technique in differentiation, becomes very handy.

Exponential Function

The exponential function, often written as \(e^x\), is a special and important mathematical function characterized by its base, the mathematical constant \(e\), approximately equal to 2.718. One of the notable features of the exponential function is that its rate of growth is proportional to its current value, which makes it grow increasingly faster. Exponential functions model various real-world phenomena such as population growth, radioactive decay, and interest calculations over time.

Key features of the exponential function:

• The derivative of \(e^x\) is unique because it is the same as the function itself, \(\frac{d}{dx}[e^x] = e^x\).
• The exponential function is always positive and never equals zero.
• It crosses the y-axis at 1 when \(x = 0\).

In our exercise, the function \(f(x) = xe^x\) involves multiplying the linear function \(x\) with the exponential function \(e^x\). When differentiating this product, recognizing that one component is exponential is crucial to apply the product rule correctly.

Calculus

Calculus is the branch of mathematics that studies continuous change. It is considered a pillar of modern mathematics and science, especially invaluable in fields such as physics, engineering, and economics. The two main concepts in calculus are differentiation, which we covered earlier, and integration.

Important concepts within calculus:

• Differentiation: As previously mentioned, this deals with calculating the rate at which quantities change.
• Integration: This is the reverse process of differentiation, used to find the total or accumulated quantity, such as area under a curve.
• Calculus enables the calculation of limits, derivatives, integrals, and infinite series.

In this exercise, calculus was central as it involves using the differentiation process and the product rule to find the function's derivative. Understanding these foundational aspects of calculus is essential to solve more complex problems and models in various scientific disciplines.