91Ó°ÊÓ

The product rule is a fundamental tool in calculus used when differentiating functions that are the product of two other functions. When you have a function of the form \( u(x) \cdot v(x) \), where both \( u \) and \( v \) are differentiable, the derivative of the product is calculated using the formula:

• \( (uv)' = u'v + uv' \)

This means that you take the derivative of the first function \( u \) times the second function \( v \), plus the first function \( u \) times the derivative of the second function \( v \).
In the exercise, this rule was applied to differentiate the term \( x^2 \ln x \). Here, \( u = x^2 \) and \( v = \ln x \), with \( u' = 2x \) and \( v' = \frac{1}{x} \). By applying the product rule, the derivative term becomes:

• \( 2x \ln x + x \)

The chain rule is another essential calculus tool used when differentiating composite functions. This rule is applicable when a function is nested inside another function, such that you have a function \( f(g(x)) \). The chain rule provides a way to differentiate by taking the derivative of the outer function, evaluated at the inner function, and multiplying it by the derivative of the inner function:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

In the original exercise, the chain rule was used to differentiate \( e^{x^2} \). Here, the outer function is the exponential \( e^u \) where \( u = x^2 \), and the inner function is \( x^2 \). The derivative of the outer function \( e^u \) is itself, \( e^u \), while the derivative of the inner function \( x^2 \) is \( 2x \). Thus, by the chain rule, the derivative becomes:

• \( 2x e^{x^2} \)

Exponential functions have a unique property where the derivative of the exponential function \( e^x \) is the function itself, \( e^x \). This property is essential when dealing with growth models and other exponential scenarios.
When the exponent itself is a function, such as \( e^{g(x)} \), the derivative involves using both the chain rule and the specific property of exponential derivatives:

• The derivative of \( e^{g(x)} \) is \( e^{g(x)} \cdot g'(x) \)

In our exercise, this was applied to \( e^{x^2} \). By differentiating the exponent \( x^2 \), which yields \( 2x \), we could then express the derivative as \( 2x e^{x^2} \), confirming the exponential function property and chain rule are combined here.

The power rule is a basic derivative rule in calculus which simplifies finding the derivative of polynomial expressions. This rule states that for any function in the form \( x^n \), its derivative is:

• \( \frac{d}{dx}[x^n] = n x^{n-1} \)

This means you multiply the power by the coefficient, and then reduce the power by one.
In the provided problem, the power rule was applied to differentiate terms like \( x^2 \) and \( -\frac{1}{2}x^2 \). For \( -\frac{1}{2}x^2 \), using the power rule gives a derivative of \( -x \). This illustrates how power functions can be simply and directly differentiated using this rule.