Understanding the Power Rule | Derivatives of Functions in the Form f(x) = x^n

d/dx x^n

The notation “d/dx” represents the derivative with respect to the variable x

The notation “d/dx” represents the derivative with respect to the variable x. 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 you have a function in the form f(x) = x^n, where n is a constant, then the derivative with respect to x is given by:

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

Let’s go through an example to illustrate this.

Example:
Find the derivative of f(x) = x^3.

Using the power rule, we differentiate the function by multiplying the exponent (3) by the coefficient (1) and then subtracting 1 from the exponent. This gives us:

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

Simplifying:

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

So, the derivative of x^3 with respect to x is 3*x^2.

In general, the power rule allows us to find the derivative of any function in the form f(x) = x^n, where n is a constant. The derivative will be n times x raised to the power of (n-1).

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