The Power Rule: Understanding Differentiation of Functions f(x) = x^n

Power Rule

The Power Rule is a fundamental rule in calculus that allows us to differentiate functions of the form f(x) = x^n, where n is any real number

The Power Rule is a fundamental rule in calculus that allows us to differentiate functions of the form f(x) = x^n, where n is any real number. Differentiation is the process of finding the derivative of a function, which tells us how the function is changing at each point.

The Power Rule states that if we have a function f(x) = x^n, then its derivative is given by f'(x) = nx^(n-1).

To understand why this rule works, let’s go over the steps for differentiating a function using the Power Rule:

1. Start with the function f(x) = x^n.
2. Rewrite the function using exponential notation: f(x) = e^(n * ln(x)). Here, e is the base of the natural logarithm and ln(x) represents the natural logarithm of x.
3. Apply the chain rule. The chain rule allows us to differentiate composite functions. In this case, we have the function g(u) = e^u, where u = n * ln(x). The derivative of g(u) is g'(u) = e^u.
4. Apply the chain rule again to differentiate u = n * ln(x). The derivative of u with respect to x is du/dx = n/x.
5. Multiply the derivatives obtained in steps 3 and 4: g'(u) * du/dx = e^u * (n/x).
6. Substitute u = n * ln(x) back into the result obtained in step 5: e^(n * ln(x)) * (n/x).
7. Simplify using the properties of exponents. Remember that e^(ln(x)) equals x: x^n * (n/x).
8. Simplify further by canceling out the common factor of x: nx^(n-1).

Hence, we have successfully obtained the derivative of the function f(x) = x^n, which is f'(x) = nx^(n-1).

It is important to note that the Power Rule is not applicable when n is zero. For n = 0, the derivative of f(x) = x^0 is f'(x) = 0.

More Answers: