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The product rule is a fundamental concept in calculus used to differentiate functions that are the product of two sub-functions. When you have a function in the form of \(u(x) \cdot v(x)\), the derivative \((uv)'\) is given by the formula:

• Take the derivative of the first function \(u'\) while leaving the second function \(v\) as is.
• Then, take the derivative of the second function \(v'\) while leaving the first function \(u\) as is.
• Add these two results together to obtain the derivative of the product.

So, the rule can be expressed as: \[(uv)' = u'v + uv'\] This allows us to efficiently find derivatives of more complex functions like \(f(x) = x^4 e^x\), by breaking them down into simpler parts.

Differentiation is the process in calculus used to find the rate at which a function is changing at any given point. This rate of change is called the derivative, and it can provide a lot of insight into the behavior of the function.The differentiation process involves applying rules like the power rule, product rule, chain rule, and others to functions. Each rule helps to handle specific types of function components effectively.For example, when differentiating \(f(x) = x^4 e^x\), we apply the product rule due to the product of two functions \(x^4\) and \(e^x\). Understanding these rules is essential for calculus, as they form the building blocks for solving more complex problems.

The first derivative of a function, denoted as \(f'(x)\), represents the instantaneous rate of change of the function. In simple terms, it shows the slope of the tangent to the function at any single point. This is useful for finding critical points, and understanding where the function is increasing or decreasing.In our example function \(f(x) = x^4 e^x\), we use the product rule to find:1. Differentiate \(x^4\) to get \(4x^3\).2. Differentiate \(e^x\) to get \(e^x\).3. Apply the product rule: - \(u'v = 4x^3 e^x\) - \(uv' = x^4 e^x\)4. Add these to get the first derivative: - \(f'(x) = 4x^3 e^x + x^4 e^x\) This expression helps analyze the behavior of the function for any input \(x\).

The second derivative \(f''(x)\) provides information about the concavity of the function and the rate of change of the first derivative. It answers questions like "Is the function curving upwards or downwards?"To find the second derivative, you differentiate the first derivative \(f'(x)\). Using the product rule again:1. Differentiate the first term \(4x^3 e^x\): - Use product rule: \((4x^3)' \, e^x + 4x^3 (e^x)' = 12x^2 e^x + 4x^3 e^x\)2. Differentiate the second term \(x^4 e^x\): - Use product rule: \((x^4)' \, e^x + x^4 (e^x)' = 4x^3 e^x + x^4 e^x\)3. Add the differentiated terms: - Combine and simplify: \(f''(x) = 12x^2 e^x + 8x^3 e^x + x^4 e^x\)This offers more in-depth insight into how the original function changes behavior, and can guide further analysis on curvature and acceleration of the function with respect to \(x\).