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An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In our function, the exponential part is represented by \( e^x \). This base \( e \) is an irrational number approximately equal to 2.718. It's called Euler's number, commonly used because it has unique properties, especially in calculus. The beauty of exponential functions like \( e^x \) is that their derivative is the same as the function itself. So, when we differentiate \( e^x \), the result is still \( e^x \). This simplicity is one reason exponential functions are prominent in many fields, including finance and the natural sciences. These functions typically model growth processes, such as population growth, radioactive decay, and compound interest, due to their continuous and proportional growth characteristics.

The product rule is a fundamental technique in calculus used to differentiate products of two functions. It enables us to tackle problems where two dependent variables need differentiation simultaneously. The rule is stated mathematically as follows: If you have two differentiable functions \( u(x) \) and \( v(x) \), then the derivative of their product \( y = u(x)v(x) \) is given by:

• \( \frac{d}{dx}[u(x) \cdot v(x)] = u(x) \cdot \frac{dv}{dx} + v(x) \cdot \frac{du}{dx} \).

In our scenario, we identified \( u(x) = e^x \) and \( v(x) = \sqrt{x^3} \). By employing the product rule, we are seamlessly able to find the derivative of these two intersecting functions. This rule is handy when solving problems involving the multiplication of different working components, making it a cornerstone concept for many calculus operations.

Radical functions are expressions involving roots, such as square roots or higher, of a variable. Our function \( v(x) = \sqrt{x^3} \) is a classic example, specifically a square root. It's often helpful to rewrite radicals using exponents to simplify differentiation and integration. For instance, \( \sqrt{x^3} \) can be rewritten as \( x^{3/2} \), which is more intuitive for calculus operations.

• The derivative of a radical function follows the power rule: if \( v(x) = x^n \), its derivative, \( \frac{dv}{dx} = nx^{n-1} \).

In this exercise, differentiating \( x^{3/2} \) gives \( \frac{3}{2}x^{1/2} \). Radical functions often appear in scenarios involving distance or area calculations, and converting them to power functions facilitates easier calculus operations.