Power Rule
Power Rule
The power rule is a fundamental principle in calculus used to differentiate functions of the form \( f(x) = x^n \), where \( n \) is a real number. This rule simplifies the process of finding the derivative of power functions and is a cornerstone in the study of differential calculus.
Definition
The power rule states that if \( f(x) = x^n \), then the derivative of \( f \) with respect to \( x \) is given by:
\[ f'(x) = nx^{n-1} \]
This rule applies to any real number \( n \), including positive integers, negative integers, and non-integer values.
Derivation
The power rule can be derived using the limit definition of the derivative. For \( f(x) = x^n \), the derivative \( f'(x) \) is defined as:
\[ f'(x) = \lim_Template:H \to 0 \frac{(x+h)^n - x^n}{h} \]
Expanding \( (x+h)^n \) using the binomial theorem and simplifying the expression leads to:
\[ f'(x) = \lim_Template:H \to 0 \frac{nx^{n-1}h + \text{higher order terms in } h}{h} \]
As \( h \) approaches 0, the higher-order terms in \( h \) vanish, leaving:
\[ f'(x) = nx^{n-1} \]
Applications
The power rule is widely used in various fields such as physics, engineering, and economics. It is essential for solving problems involving rates of change, optimization, and modeling natural phenomena.
Physics
In physics, the power rule is used to find the velocity and acceleration of objects. For example, if the position of an object is given by \( s(t) = t^3 \), the velocity \( v(t) \) is the first derivative of \( s(t) \):
\[ v(t) = \frac{d}{dt} t^3 = 3t^2 \]
The acceleration \( a(t) \) is the derivative of the velocity:
\[ a(t) = \frac{d}{dt} 3t^2 = 6t \]
Engineering
In engineering, the power rule helps in analyzing the behavior of electrical circuits, mechanical systems, and fluid dynamics. For instance, in electrical engineering, the power dissipated in a resistor is given by \( P = I^2R \), where \( I \) is the current and \( R \) is the resistance. The rate of change of power with respect to current can be found using the power rule:
\[ \frac{dP}{dI} = \frac{d}{dI} (I^2R) = 2IR \]
Economics
In economics, the power rule is used to determine marginal cost and revenue functions. If the total cost \( C(x) \) of producing \( x \) units is given by \( C(x) = x^4 \), the marginal cost \( MC(x) \) is the derivative of \( C(x) \):
\[ MC(x) = \frac{d}{dx} x^4 = 4x^3 \]
Generalizations
The power rule can be extended to more complex functions through the use of the chain rule and the product rule. For example, if \( f(x) = (g(x))^n \), where \( g(x) \) is a differentiable function, the derivative is:
\[ f'(x) = n(g(x))^{n-1} g'(x) \]
This is a direct application of the chain rule.
Historical Context
The power rule was first formulated by Isaac Newton and Gottfried Wilhelm Leibniz independently in the late 17th century. Their work laid the foundation for modern calculus and significantly advanced the study of mathematics and science.
Examples
Positive Integer Powers
For \( f(x) = x^5 \):
\[ f'(x) = 5x^4 \]
Negative Integer Powers
For \( f(x) = x^{-3} \):
\[ f'(x) = -3x^{-4} \]
Fractional Powers
For \( f(x) = x^{1/2} \):
\[ f'(x) = \frac{1}{2}x^{-1/2} \]
Irrational Powers
For \( f(x) = x^{\pi} \):
\[ f'(x) = \pi x^{\pi-1} \]
Limitations
The power rule is not applicable to functions that are not differentiable. For example, the function \( f(x) = |x| \) is not differentiable at \( x = 0 \). Additionally, the power rule does not apply to functions with complex exponents without further modifications.