The Power Rule for Finding Derivatives of x Raised to the Power of n

d/dx (x^n)

The derivative of x raised to the power of n, denoted as d/dx (x^n), can be found using the power rule of derivatives

The derivative of x raised to the power of n, denoted as d/dx (x^n), can be found using the power rule of derivatives. The power rule states that if we have a function of the form f(x) = x^n, then the derivative can be obtained by multiplying the exponent n by the coefficient in front of x, and then decreasing the exponent by 1.

So, for d/dx (x^n), we would have:

d/dx (x^n) = n * x^(n-1)

In simpler terms, we take the exponent n and multiply it by the coefficient 1 (since x is equivalent to x^1), and then decrease the exponent by 1 to get x^(n-1).

For example, if we have the function f(x) = x^3, then the derivative is:

d/dx (x^3) = 3 * x^(3-1) = 3 * x^2

On the other hand, if we have f(x) = x^(-2), then the derivative is:

d/dx (x^(-2)) = -2 * x^(-2-1) = -2 * x^(-3) = -2/x^3

In summary, the derivative of x raised to any power n can be obtained by multiplying n by x^(n-1).

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