91Ó°ÊÓ

The Power Rule is a fundamental tool in calculus for differentiation. It's used when you have a function in the form of a power, like \( f(x) = x^n \). The rule states that the derivative of \( x^n \) is \( nx^{n-1} \). In simpler terms, you multiply by the power, then subtract one from the power.
To illustrate, if you have a function \( y = x^3 \), applying the Power Rule gives you \( y' = 3x^{2} \). This makes finding the rate of change quick and efficient.

• Identify the power of the expression.
• Multiply by the exponent.
• Subtract one from the exponent.

The Power Rule is especially useful when dealing with functions that have exponents, particularly when these functions are part of more complex operations, as in the given exercise. Here, the phrase "apply the Power Rule" helps us recognize how straightforward it can be to handle powers in differentiation.

The Chain Rule is used when differentiating a "function of a function," meaning one that involves a combination of two functions. If you see something like \( f(x) = g(h(x)) \), the Chain Rule helps break it down. It's expressed as \( f'(x) = g'(h(x)) \cdot h'(x) \). So, you differentiate the outer function first and then multiply by the derivative of the inner function.
An example would be if \( f(x) = (2x+3)^2 \), you'd differentiate the outer function \((u^2)\) to get \( 2u \), and then the inner function \((2x+3) \to 2 \), resulting in \( 4(2x+3) \).

• Identify the outer and inner functions.
• Dfferentiate the outer function, leaving the inner function intact.
• Multiply by the derivative of the inner function.

This approach simplifies the process of tackling complex functions, especially when they involve composite structures like nested logarithms or other expressions. In this exercise, after rewriting the function to \((\ln x)^{1/5}\), the Chain Rule was crucial for handling the nested nature of these operations.

Differentiation of logarithmic functions, specifically the natural log, which is denoted as \( \ln(x) \), follows a straightforward rule. The derivative of \( \ln(x) \) is \( \frac{1}{x} \). This makes logarithmic differentiation particularly useful as it simplifies complex algebraic expressions, breaking them down into manageable parts.
For example, if your function is \( y = \ln(x) \), its derivative directly becomes \( y' = \frac{1}{x} \). Another practical application is when you have a logarithm raised to a power or multiplied within another function.

• Notice how the derivative converts a logarithm into a reciprocal.
• Use it in combination with other differentiation rules like the Chain Rule.

In the given exercise with the function \( \ln(x) \), understanding its simple derivative is essential when applying more advanced rules such as the Chain and Power Rule together. This makes simplification of derivatives accessible and manageable.