91Ó°ÊÓ

The power rule is a fundamental technique in calculus for finding derivatives of polynomial functions. It is especially useful when dealing with functions of the form \( f(x) = x^n \), where \( n \) is any real number. The rule states that the derivative \( f'(x) \) is found by multiplying \( n \) by \( x \) raised to the power of \( n-1 \).
For instance, if \( f(x) = x^2 \), applying the power rule gives \( f'(x) = 2x^{2-1} = 2x \).
This simplifies the process of finding derivatives significantly, especially as functions grow more complex.

Applying Power Rule:

• Identify the power \( n \) of \( x \).
• Multiply the expression by the power \( n \).
• Decrease the power by 1.
• Write the simplified expression as the derivative.

It is crucial to use the power rule correctly when it's applicable, as it forms the basis for more advanced derivative techniques.

The chain rule is a powerful tool in calculus, used to differentiate composite functions. Composite functions are those where one function is nested within another, such as \( h(x) = f(g(x)) \). To differentiate such functions, the chain rule provides a systematic way to do so.
The essence of the chain rule can be captured with the equation \( h'(x) = f'(g(x))g'(x) \).

Steps to Using the Chain Rule:

• Differentiate the outer function while keeping the inner function intact.
• Multiply by the derivative of the inner function.

Let's say \( g(x) = e^{-x} \). Here \( h(u) = e^u \) where \( u(x) = -x \). The outer function \( e^u \) differentiates to \( e^u \) and the derivative of \( u(x) \) would be \( -1 \), thus leading to \( -e^{-x} \) for \( g'(x) \).
The chain rule is critical as it allows us to tackle more complicated functions by breaking them into manageable parts.

When faced with differentiating products of functions, the product rule becomes indispensable. If you have two functions, \( f(x) \) and \( g(x) \), that are multiplied together, the product rule provides a method to find the derivative of their product directly.
The formula for the product rule is \( (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) \).

Applying the Product Rule:

• Find the derivative of the first function \( f'(x) \).
• Multiply it by the second function \( g(x) \).
• Find the derivative of the second function \( g'(x) \).
• Multiply it by the first function \( f(x) \).
• Add these two products together to get the final derivative.

In our exercise, multiplying the derivative of \( x \) which is 1, by \( e^{-x} \), and adding \( x \) times the derivative of \( e^{-x} \) which is \(-e^{-x}\) provides the full derivative: \( e^{-x} - x e^{-x} \).
The product rule complements the power and chain rules, providing a complete toolkit for differentiating a wide range of functions.