Problem 1 - Single Variable Calculus

Question

Use the Product Rule to find the derivative of each function. \(f(x)=x^{2} e^{x}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=x^2e^x\) using the Product Rule is given by: \(f'(x) = x e^x(2 + x)\).

Step 1: 1. Identify the functions#g(x)# and #h(x)

In our given function \(f(x)=x^{2}e^{x}\), we can identify \(g(x)=x^2\) and \(h(x)=e^x\).

Step 2: 2. Differentiate #g(x)# with respect to #x

We differentiate \(g(x)=x^2\) with respect to \(x\) using the Power Rule: \(g'(x) = 2x\).

Step 3: 3. Differentiate #h(x)# with respect to #x

The derivative of \(h(x)=e^x\) with respect to \(x\) is simply \(h'(x) = e^x\).

Step 4: 4. Apply the Product Rule to find #f'(x)

Now we apply the Product Rule to find the derivative of the function \(f(x)=x^2e^x\): \[f'(x) = g'(x)h(x) + g(x)h'(x)\] Substituting the derivatives and functions we found in steps 2 and 3: \[f'(x) = (2x)(e^x) + (x^2)(e^x) \]

Step 5: 5. Simplify the derivative if possible

We can simplify the expression for \(f'(x)\) by factoring out the common factor \(x e^x\): \[f'(x) = x e^x(2 + x)\] So, the derivative of the function \(f(x)=x^2e^x\) is: \[f'(x) = x e^x(2 + x)\]

Key Concepts

Derivative of Exponential Functions

Understanding the derivative of exponential functions is crucial for calculus, as these functions frequently occur in various applications, from compound interest problems to population growth studies. In mathematical terms, if you have a function of the form \(f(x) = e^{g(x)}\), where \(e\) is the base of the natural logarithm, the derivative is given by \(f'(x) = e^{g(x)}g'(x)\). This simply states that the derivative of an exponential function is the original function multiplied by the derivative of the exponent.

Let's apply this to a real-world scenario. Suppose you're looking into how a population \(P\) changes over time. If \(P\) is modeled by \(P(t) = e^{rt}\), where \(t\) represents time and \(r\) is a constant rate of growth, then the rate of change of the population would be \(P'(t) = re^{rt}\). This insight allows biologists and demographers to understand and predict changes in population sizes over time.

Power Rule in Calculus

One of the most important and widely used differentiation rules in calculus is the power rule. This rule is applied when differentiating polynomials, which are functions composed of terms like \(x^n\), where \(n\) is any real number exponent. The rule is simple: for any function \(g(x) = x^n\), its derivative \(g'(x)\) is \(nx^{n-1}\).

Imagine you're interested in the concept of velocity in physics, which essentially involves differentiation. If an object is moving along a path and its position at time \(t\) is given by \(s(t) = t^3\), then according to the power rule, its velocity \(v(t)\) would be the derivative of \(s(t)\), which is \(v(t) = 3t^2\). This illustrates how the power rule is used to find rates of change in real-life scenarios.

Simplifying Derivatives

Simplifying derivatives is a key step in calculus that helps us make sense of the results we obtain after differentiation. By factoring, combining like terms, or canceling common factors, you can make complicated expressions more manageable. When simplifying, always look for common factors and terms that can be combined. This not only makes the expressions easier to work with but also enables a clearer understanding of the function's behavior.

For instance, when dealing with economic functions such as a cost function \(C(x)\) that includes terms with the same base raised to an exponent, simplifying can help in figuring out marginal costs and in making critical business decisions regarding production levels and pricing.

Calculus Problem Solving

Effective problem solving in calculus often requires a step-by-step approach which encompasses understanding the problem, identifying the appropriate rules, performing the calculations, and simplifying the results. This process also involves a combination of analytical skills and a solid grasp of calculus concepts.

In applied fields, such as engineering, this systematic approach is used in designing components. An engineer might analyze the stress \(\sigma(x)\) on a beam under load, which might be a function involving various rates of change. Through differentiation and solving the resulting equations, the engineer can understand how stress varies along the beam and, ultimately, ensure the safety and efficiency of their designs.