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This tutorial covers the power rule, an essential rule in calculus for differentiating functions of the form x^n, where n is any real number.
What is the Power Rule?
The power rule is a derivative rule that applies to functions of the form f(x) = x^n, where n is a real number.
Power Rule states that the derivative of x^n with respect to x is nx^(n-1).
When put in a formula,
(d/dx) x^n == n*x^(n-1)
where n is a real number.
Examples of the Power Rule
1. Find the derivative of the function f(x) = x^3.
Solution
Given: f(x) = x^3
Find: We need to find the derivative of f(x).
Applying Formula: Since f(x) = x^3 is in the form of x raised to the power of n.
Let us apply power rule.
(d/dx) x^n == n*x^(n-1)
From given function f(x) = x^3, n = 3.
(d/dx) x^3 == 3*x^(3-1)
(d/dx) x^3 == 3*x^(2)
Therefore,
(d/dx) x^3 == 3*x^(2)
2. Find the derivative of the function f(x) = x^-4.
Solution
Given: f(x) = x^-4
Find: We need to find the derivative of f(x).
Applying Formula: Since f(x) = x^-4 is in the form of x raised to the power of n.
Let us apply power rule.
(d/dx) x^n == n*x^(n-1)
From given function f(x) = x^-4, n = -4.
(d/dx) x^-4 == -4*x^(-4-1)
(d/dx) x^-4 == -4*x^(-5)
Therefore,
(d/dx) x^-4 == -4*x^(-5)