Problem 19 - Applied Calculus

Question

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(w)=\left(5 w^{2}+3\right) e^{w^{2}} $$

Step-by-Step Solution

Answer

The derivative is \(f'(w) = (10w^3 + 16w) e^{w^2}\).

Step 1: Identify the Function

The function given is a product of two expressions: \(f(w) = (5w^2 + 3) e^{w^2}\).It consists of two parts: a polynomial \(5w^2 + 3\) and an exponential function \(e^{w^2}\).

Step 2: Apply the Product Rule

To differentiate the product of two functions, use the product rule:If \(u(w) = 5w^2 + 3\) and \(v(w) = e^{w^2}\), then by the product rule:\(f'(w) = u'(w)v(w) + u(w)v'(w)\).

Step 3: Differentiate the Polynomial Part

Differentiate the polynomial part \(u(w) = 5w^2 + 3\):\(u'(w) = \frac{d}{dw}(5w^2 + 3) = 10w\).

Step 4: Differentiate the Exponential Part

Differentiate the exponential part \(v(w) = e^{w^2}\) using the chain rule:\(v'(w) = \frac{d}{dw}(e^{w^2}) = e^{w^2} \cdot \frac{d}{dw}(w^2) = 2we^{w^2}\).

Step 5: Combine Results Using Product Rule

Substitute \(u'(w)\), \(v(w)\), \(u(w)\), and \(v'(w)\) into the product rule formula:\(f'(w) = (10w) \cdot e^{w^2} + (5w^2 + 3) \cdot (2w e^{w^2})\).

Step 6: Simplify the Expression

Expand and simplify the expression:\[f'(w) = 10w e^{w^2} + (5w^2 + 3) \cdot 2w e^{w^2}\]\[= 10w e^{w^2} + (10w^3 + 6w) e^{w^2}\]Combine like terms:\[= (10w + 10w^3 + 6w) e^{w^2}\]Simplify further:\[= (10w^3 + 16w) e^{w^2}\].

Key Concepts

Product Rule

In calculus, the product rule is an essential technique for finding the derivative of a product of two functions. The product of two functions means that these functions are multiplied together. If you have two functions, say \(u(w)\) and \(v(w)\), the product rule states that the derivative of their product \(u(w) \cdot v(w)\) is:\[ \frac{d}{dw}[u(w) \cdot v(w)] = u'(w) \cdot v(w) + u(w) \cdot v'(w) \] This rule is useful in situations where two more complex functions combine into one, allowing you to handle each function separately and then combine the results according to the rule.
In this exercise, the function consists of a polynomial \(5w^2 + 3\) and an exponential function \(e^{w^2}\). Using the product rule means that we take the derivative of each part, separately, and then carefully combine them as described above.

Chain Rule

The chain rule is another crucial differentiation technique, particularly when dealing with composite functions. A composite function means that one function is inside another function, like nesting. If you have a composition of functions such as \(f(g(x))\), the chain rule provides a means to differentiate through the different layers of the functions. It states:

• First, differentiate the outer function, leaving the inner function unchanged.
• Second, multiply by the derivative of the inner function.
• Mathematically, it looks like: \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

In our example, we have an exponential function \(e^{w^2}\), where \(w^2\) is the inner function. By the chain rule, the differentiation process becomes: \( e^{w^2} \times \frac{d}{dw}(w^2) \), which follows to \(2w e^{w^2}\). This method allows you to efficiently unwrap layered functions step-by-step.

Polynomial Differentiation

Polynomial differentiation is one of the more straightforward differentiation processes, due to its repetitive nature following well-defined rules. Polynomials are expressions consisting of variables and coefficients that include terms in the form of powers. The basic rule for differentiating polynomials \(ax^n\) is to multiply the coefficient \(a\) by the power \(n\), then reduce the power by one:\[ \frac{d}{dx}[ax^n] = a \cdot n \cdot x^{n-1} \]
For instance, in our function, where we have the polynomial \(5w^2 + 3\), the derivative comes directly from applying this formula to each term. The term \(5w^2\) becomes \(10w\) after differentiation, and the constant \(3\) differentiates to \(0\), as constants drop out of the differentiation process. Understanding polynomial differentiation is foundational for tackling more complex functions.

Exponential Function Differentiation

Exponential functions often involve expressions where the variable appears in the exponent, such as \(e^{x}\). Differentiating these functions leveraged two simple but powerful rules:

• An exponential function retains its form when differentiated, meaning that \( \frac{d}{dx}[e^{x}] = e^{x}\).
• If the function is a composite, like \(e^{g(x)}\), you'll need to apply the chain rule.

This is precisely what happens in the provided exercise with \(e^{w^2}\). The differentiation first acknowledges the constant nature of the exponential part \(e^{...}\), and then, importantly, applies the chain rule to \(w^2\), giving \(2w e^{w^2}\).
Exponential and polynomial differentiation often go hand in hand, combining the rules effectively to handle complex expressions.