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When you have a function that involves two other functions being multiplied together, you'll often use the product rule to find its derivative. In our case, the function is a product of two separate functions: \( u(w) = 5w^2 + 3 \) and \( v(w) = e^{w^2} \). The product rule provides a formula to differentiate such products:

• The formula is: \((uv)' = u'v + uv'\).

This formula basically tells us to take the derivative of the first function \( u' \), multiply it by the original second function \( v \), and then add it to the product of the original first function \( u \) and the derivative of the second function \( v' \). You'll follow this formula whenever you come across a function that is clearly a product of two other functions. Remembering the product rule will greatly simplify working with and differentiating products of functions. Find each derivative separately, then plug them into the formula.

The chain rule is essential for finding the derivative of a composite function, which is a function within another function. In the given exercise, to differentiate \( v(w) = e^{w^2} \), we need to apply the chain rule because \( e^{w^2} \) is technically a function (exponential) of another function \( w^2 \).Here's how the chain rule works:

• Identify the outer function and the inner function. Here, the outer function is \( e^g \) and the inner function is \( g(w) = w^2 \).
• First, differentiate the outer function while keeping the inner function intact. This gives us \( e^{g(w)} \).
• Then, multiply by the derivative of the inner function \( g'(w) = 2w \).

So for our problem, we find that the chain rule gives us the derivative of \( v(w) = e^{w^2} \) as \( v'(w) = 2we^{w^2} \). Using the chain rule is a vital skill, especially when dealing with functions within functions, and it simplifies what could be very complex differentiation steps.

Breaking down the differentiation into clear steps is crucial for ensuring accuracy. Let's look at the differentiation process for our function \( f(w) = (5w^2 + 3)e^{w^2} \):- **Step 1:** **Identify the type of function** you are dealing with. Recognize that it is a product of two functions and decide to use the product rule.- **Step 2:** **Differentiate each component function** separately. First, \( u(w) = 5w^2 + 3 \), becomes \( u'(w) = 10w \). Then, apply the chain rule to \( v(w) = e^{w^2} \) to get \( v'(w) = 2we^{w^2} \).- **Step 3:** **Apply the product rule formula.** Plug each of the derivatives into the formula, getting: \[ f'(w) = (10w)e^{w^2} + (5w^2 + 3)(2we^{w^2}) \].- **Step 4:** **Simplify the expression.** Expand and combine like terms: \[ f'(w) = (10w^3 + 16w)e^{w^2} \].By dividing your differentiation into these steps, you ensure the process is logical and systematic, reducing the risk of errors. Practice this approach to enhance your calculus proficiency.