Question

Find the derivative of each function y=x^(8)

Answer 1

The derivative of the function
`\(y = x^8\)` is
`\(y' = 8x^7\).`

Step-by-step explanation:

To find the derivative of the given function
`\(y = x^8\),` we can apply the power rule of differentiation. The power rule states that if
`\(f(x) = x^n\),` then
`\(f'(x) = nx^((n-1))\)` . In this case, the exponent is
`\(8\),` and applying the power rule yields the derivative
`\(y' = 8x^7\).`

Understanding differentiation rules is fundamental in calculus, as it allows us to find the rate at which a function changes. The power rule is a specific case of these rules and is particularly useful when dealing with functions involving powers of
`\(x\)` . In the context of
`\(y = x^8\` ), the derivative
`\(y'\)` provides the instantaneous rate of change of
`\(y\)` with respect to
`\(x\)` at any given point.

The result
`\(8x^7\)` signifies that as
`\(x\)` varies, the rate at which
`\(y\)` changes is proportional to
`\(x^7\)` . This exponential relationship is a key characteristic of power functions. The exponent
`\(7\)` in the derivative indicates that the rate of change increases rapidly as
`\(x\)` increases, showcasing the behavior of higher-degree power functions.

Derivatives play a crucial role in analyzing functions and their behavior. They provide insights into the slope, concavity, and rate of change, allowing mathematicians and scientists to model and understand a wide range of phenomena in various fields.