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Magazine Chapter 13: Problem 61
Find \(f^{\prime}(x)\) using the alternative definition. $$f(x)=x^{3}$$
Short Answer
Expert verified
The derivative \(f'(x) = 3x^2\).
Step by step solution
01
Write the Alternative Definition of the Derivative
The alternative definition of the derivative, also known as the limit definition, is given by: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]. Our task is to find this limit expression for the given function \( f(x) = x^3 \).
02
Substitute into the Definition
Substitute \( f(x+h) \) and \( f(x) \) into the alternative definition: \[ f'(x) = \lim_{h \to 0} \frac{(x+h)^3 - x^3}{h} \].
03
Expand \((x+h)^3\)
Use the binomial theorem to expand \((x+h)^3\): \[ (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \].
04
Simplify the Expression
Substitute the expansion into the expression: \[ f'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} \]. Cancel \(x^3\) terms: \[ f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} \].
05
Factor the Expression
Factor a \(h\) out of the numerator: \[ f'(x) = \lim_{h \to 0} \frac{h(3x^2 + 3xh + h^2)}{h} \]. Cancel \(h\) from the numerator and the denominator: \[ f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2) \].
06
Evaluate the Limit
Evaluate the limit as \(h \to 0\): \[ f'(x) = 3x^2 + 3x(0) + (0)^2 \]. Simplifying gives: \[ f'(x) = 3x^2 \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Derivatives
In calculus, derivatives represent how a function changes as its input changes. Think of the derivative as the "slope" of the function at any given point. It describes the rate of change of a quantity, which can be a powerful tool in both math and real-world applications.
- The derivative of a function at a point gives the instantaneous rate of change of that function at that specific point.
- It is symbolized by \( f'(x) \) or \( \frac{df}{dx} \), where \( f(x) \) is the function.
Limit Definition of a Derivative
The limit definition is the most fundamental way to find a derivative and involves calculating the limit of the difference quotient. This rigorous foundation helps understand why derivatives work as they do.To find the derivative of a function \( f(x) \) using the limit definition:
- Consider the function \( f(x) \) and its slight change \( f(x+h) \) when the input increases by \( h \).
- The limit definition is expressed as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
- This formula calculates the slope of the tangent line to the curve at point \( x \).
- The \( h \) in the denominator ensures that the difference becomes infinitesimally small, capturing the exact slope at a point.
Expanding Polynomials: Binomial Theorem
The binomial theorem is a key tool for expanding expressions raised to a power. Understanding this theorem is crucial when deriving functions, especially when using the limit definition of the derivative.The binomial theorem states:
- When expanding \((x+h)^n\), each term follows the pattern: \[ \sum_{k=0}^{n} \binom{n}{k} x^{n-k} h^{k} \]
- The term \( \binom{n}{k} \) stands for "n choose k" and calculates the number of ways to choose \( k \) elements from the set of \( n \) elements. This is given by \( \frac{n!}{k!(n-k)!} \).
- For small \( h \), higher powers of \( h \) become insignificant in the derivative calculation due to the limit \( h \to 0 \).
