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The product rule is a valuable tool in calculus for differentiating functions that are a product of two or more simpler functions. If you have two functions, say \( u(w) \) and \( v(w) \), both depending on \( w \), the derivative of their product \( u(w) v(w) \) with respect to \( w \) is given by:

• \( (uv)' = u'v + uv' \)

This means you take the derivative of the first function \( u \) while leaving \( v \) undifferentiated, and then add the product of the undifferentiated \( u \) with the derivative of \( v \).
In the exercise, \( h(w) = w \arcsin w \), we let \( u = w \) and \( v = \arcsin w \), where \( u' = 1 \) and \( v' = \frac{1}{\sqrt{1-w^2}} \).

So, by the product rule, the derivative \( h'(w) \) becomes \( u'v + uv' = \arcsin w + \frac{w}{\sqrt{1-w^2}} \).
This step-by-step approach helps to ensure all parts of the function are appropriately differentiated.

The arcsin function is the inverse of the sine function, usually denoted as \( \arcsin w \). It means given a sine value \( w \), the function returns the angle \( \theta \) within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
When differentiating the arcsin function with respect to \( w \), this results in:

• \( \frac{d}{dw}(\arcsin w) = \frac{1}{\sqrt{1-w^2}} \)

This derivative formula emerges because of the trigonometry properties and limits applied to \( \arcsin \).
It is important to remember that \( -1 \leq w \leq 1 \), since outside this domain the arcsin function is not defined in the real numbers.
The derivative \( \frac{1}{\sqrt{1-w^2}} \) shows how sensitive the arcsin function is to changes in \( w \). Python and other programming languages allow direct calculations for such functions.

Simplification means reducing expressions to a form that is easier to work with or interpret. In calculus, before differentiating, it can sometimes help to simplify the expression.
Although simplification might not drastically alter the way we differentiate \( h(w) = w \arcsin w \), recognizing it as a product of \( w \) and \( \arcsin w \) directs us to the correct usage of the product rule.
Steps for simplification often include:

• Factoring expressions.
• Cancelling common terms.
• Rewriting functions using known identities or relationships.

For \( h(w) \), nothing simplifies further without changing the form of the equation we set out to differentiate.
It's a gentle reminder that sometimes simplifying doesn't mean making the expression shorter; it means making it more comprehensible for the processes we intend to perform on it.

The process of differentiation involves finding how a function changes at any point, essentially measuring the rate of change. It's fundamental in calculus for analyzing the behavior of functions.
Differentiating \( h(w) = w \arcsin w \) involves applying the product rule, as these are two distinct functions multiplying each other.
Differentiation steps include:

• Identifying which rules apply, like the product rule in this case.
• Computing individual derivatives: \( \frac{d}{dw}w = 1 \) and \( \frac{d}{dw}(\arcsin w) = \frac{1}{\sqrt{1-w^2}} \).
• Combining these using the product rule to get the overall derivative.

This process is crucial in solving real-world problems where changes and trends need quantifying, like in physics for speed calculations, or in economics for cost analysis.
Understanding the roles of each function in the product and applying the differentiation rules appropriately unveils the slope or rate of change for \( h(w) \), offering insights into its behavior over its domain.