Power Rule for Derivatives
The power rule is a formula in calculus that allows us to find the derivative of a function that is in the form of a power function
The power rule is a formula in calculus that allows us to find the derivative of a function that is in the form of a power function. It states that for any real number n and a constant c, the derivative of x raised to the power of n, denoted as d/dx(x^n), is equal to n times x raised to the power of n-1.
Mathematically, if we have a function f(x) = x^n, where n is a real number, then the derivative of f(x) with respect to x is given by:
d/dx(x^n) = n x^(n-1)
Let’s go through a few examples to illustrate how this rule can be applied:
Example 1:
Consider the function f(x) = x^2. Using the power rule, we can find the derivative of this function as follows:
d/dx(x^2) = 2 x^(2-1)
= 2x
So, the derivative of f(x) = x^2 is f'(x) = 2x.
Example 2:
Let’s take the function g(x) = x^3. Applying the power rule, we can find the derivative:
d/dx(x^3) = 3 x^(3-1)
= 3x^2
Therefore, the derivative of g(x) = x^3 is g'(x) = 3x^2.
Example 3:
Consider the function h(x) = 5x^4. Using the power rule, we can find the derivative:
d/dx(5x^4) = 4 × 5 x^(4-1)
= 20x^3
So, the derivative of h(x) = 5x^4 is h'(x) = 20x^3.
It’s important to note that the power rule applies when the exponent n is a constant. If the exponent varies with x, then we need to use the chain rule. Also, the power rule is only valid for exponents that are real numbers; it does not apply to complex exponents.
More Answers:
Understanding Limit as X Approaches Infinity and Zero in MathematicsMaster the Fundamentals of Calculus: Understanding the Limit Definition of Derivative
Understanding the Limit Definition of a Derivative: A Step-by-Step Guide