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 in calculus
The derivative of x raised to the power of n, denoted as d/dx [x^n], can be found using the power rule in calculus. The power rule states that if we have a function of the form f(x) = x^n, where n is a constant, then the derivative of f(x) with respect to x is given by:
d/dx [x^n] = n * x^(n-1)
Let’s break down the process of finding the derivative of x^n using the power rule:
Step 1: Identify the function
In this case, the function is f(x) = x^n.
Step 2: Apply the power rule
To differentiate the function, we take the exponent (n) and multiply it by the coefficient in front of x (which is 1) to get n * x^(n-1).
So, the derivative of x^n with respect to x is:
d/dx [x^n] = n * x^(n-1).
For example, if we have the function f(x) = x^3, the derivative with respect to x would be:
d/dx [x^3] = 3 * x^(3-1) = 3 * x^2.
It is important to note that the power rule holds for all real numbers n, except for when n is -1 (i.e., x^(-1)). In that case, the process requires using the negative power rule, which involves negative exponents and the quotient rule.
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