The Power Rule: Finding the Derivative of x Raised to the Power of n

d/dx (x^n)

To find the derivative of x raised to the power of n, denoted as x^n, we can use the power rule

To find the derivative of x raised to the power of n, denoted as x^n, we can use the power rule. The power rule states that if we have a function of the form f(x) = x^n, then the derivative of f(x), denoted as f'(x) or df/dx, is given by:

f'(x) = n * x^(n-1)

Let’s apply the power rule to find the derivative of x^n:

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

So, the derivative of x raised to the power of n is n times x raised to the power of (n-1).

For example, if we have the function f(x) = x^3, we can find its derivative as follows:

f'(x) = 3 * x^(3-1) = 3x^2

Similarly, if we have the function g(x) = x^5, we can find its derivative as follows:

g'(x) = 5 * x^(5-1) = 5x^4

Note that the power rule is only applicable when n is a constant. If n is a function of x (for example, if we have the function f(x) = (x^2)^x), then the chain rule and logarithmic differentiation may be required to find the derivative.

More Answers: