Derivatives (Power Rule vs. Limit Definition) FAQs
What is the limit definition of a derivative?
The limit definition of a derivative is the foundational way to define the instantaneous rate of change of a function at a specific point. It represents the slope of the tangent line to the function's graph at that point.
The definition is given by:
f'(x) = limh→0 [f(x + h) - f(x)] ÷ h
Alternatively, at a specific point 'a':
f'(a) = limx→a [f(x) - f(a)] ÷ (x - a)
This definition captures the idea of finding the slope of secant lines that approach the tangent line as the interval shrinks to zero.
What is the Power Rule for derivatives?
The Power Rule is a shortcut formula derived directly from the limit definition. It provides a quick way to find the derivative of functions in the form of f(x) = xn, where 'n' is any real number.
The rule states:
If f(x) = xn, then f'(x) = n * x(n-1)
You bring the exponent 'n' down as a coefficient and reduce the original exponent by 1.
How are the Limit Definition and the Power Rule related?
The Power Rule is **derived from** the limit definition of the derivative. You can prove the Power Rule for various types of exponents (positive integers, negative integers, rational numbers) by applying the limit definition to the function f(x) = xn and evaluating the limit.
For example, to prove the power rule for f(x) = x2 using the limit definition:
f'(x) = limh→0 [(x + h)2 - x2] ÷ h
f'(x) = limh→0 [(x2 + 2xh + h2) - x2] ÷ h
f'(x) = limh→0 [2xh + h2] ÷ h
f'(x) = limh→0 [h(2x + h)] ÷ h
f'(x) = limh→0 (2x + h)
f'(x) = 2x + 0 = 2x
This matches the Power Rule result for n=2: 2 * x(2-1) = 2x1 = 2x.
When do you use the Limit Definition vs. the Power Rule?
- **Limit Definition:** You use the limit definition when you are first learning about derivatives to understand the underlying concept, or when you need to prove derivative rules (like the Power Rule itself), or sometimes for functions where simple rules don't directly apply (though other limit techniques might be involved).
- **Power Rule:** You use the Power Rule as a shortcut to quickly calculate the derivative of functions in the form of
xn(including constants, as k = k * x0) during routine differentiation problems once the rule has been established.
The limit definition explains *why* the derivative works, while the Power Rule provides a *method* for calculating it for a specific class of functions.
