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The power rule is a fundamental concept in calculus that simplifies differentiation of functions involving powers of a variable. It states that for a function of the form \( f(x) = x^n \), the derivative is given by \( f'(x) = nx^{n-1} \). This means you multiply by the exponent and reduce the exponent by 1.
For example, if you have \( y = x^4 \), according to the power rule, the derivative would be \( y' = 4x^3 \). This makes differentiating polynomial functions straightforward.
In our case, after rewriting the function \( y = \frac{1}{5x^5} \) as \( y = 5^{-1} \times x^{-5} \), we apply the power rule to \( x^{-5} \). The derivative becomes \( -5x^{-6} \), as the initial rule suggests. Understanding and applying the power rule helps immensely in simplifying complex differentiation tasks.

Function rewriting involves changing the form of the function to make it easier to work with. This is particularly useful when dealing with fractions or other expressions that complicate differentiation.
For instance, the original function \( y = \frac{1}{5x^5} \) is not in a form that is easily differentiable. By rewriting it as \( y = 5^{-1} \times x^{-5} \), it becomes clearer how to apply differentiation rules.
This step allows you to separate constants from variables and handle each part systematically. Rewriting functions is a common strategy for making complex calculus problems more manageable. It also helps in recognizing patterns that align with rules like the power rule, aiding in the differentiation process.

Simplification is the last and crucial step in solving any differentiation problem. It ensures that the expression is in its simplest form, making it easier to interpret and use in further calculations.
After differentiating the function \( y = 5^{-1} \times x^{-5} \) and obtaining \( y' = -5 \times 5^{-1} \times x^{-6} \), simplifying it to \( y' = -x^{-6} \) makes the equation neat and concise.
Simplification often involves combining like terms, reducing fractions, and eliminating unnecessary constants. It ensures that the function is presented in a generally accepted standard form, which is important for both understanding and application in more complex mathematical problems.