Derivative of 𝑥^𝑛: Understanding the Power Rule of Differentiation and its Application

𝑑/𝑑𝑥[𝑥^𝑛]

The derivative of 𝑥 raised to the power 𝑛, denoted by 𝑑/𝑑𝑥[𝑥^𝑛], can be found using the power rule of differentiation. The p

The derivative of 𝑥 raised to the power 𝑛, denoted by 𝑑/𝑑𝑥[𝑥^𝑛], can be found using the power rule of differentiation. The power rule states that when differentiating a variable raised to a constant power, you multiply the variable by the constant power, and then decrease the power by 1.

In this case, since 𝑥^𝑛 has 𝑛 as the exponent, the power rule tells us that the derivative of 𝑥^𝑛 is 𝑛 times 𝑥^(𝑛-1).

𝑑/𝑑𝑥[𝑥^𝑛] = 𝑛𝑥^(𝑛-1)

Here, the derivative 𝑑/𝑑𝑥[𝑥^𝑛] represents how the function 𝑥^𝑛 changes as 𝑥 changes.

For example, if we have the function 𝑓(𝑥) = 𝑥^3, we can find its derivative by applying the power rule. The exponent here is 3, so we multiply 3 by 𝑥^(3-1) which simplifies to 𝑥^2.

𝑑/𝑑𝑥[𝑥^3] = 3𝑥^2

The derivative of 𝑥^3 is 3𝑥^2. This means that as 𝑥 changes, the rate of change of 𝑥^3 is 3 times 𝑥^2.

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