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

User Bob Cross
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1 Answer

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Final Answer:

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.

User Oak
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