Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Find \(f^{\prime}(x)\) using the alternative definition. $$f(x)=x^{3}$$

Step-by-Step Solution

Answer

The derivative \(f'(x) = 3x^2\).

Step 1: 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 \( f(x) = x^3 \).

Step 2: Substitute into the Definition

Substitute \( f(x+h) \) and \( f(x) \) into the alternative definition: \[ f'(x) = \lim_{h \to 0} \frac{(x+h)^3 - x^3}{h} \].

Step 3: Expand \((x+h)^3\)

Use the binomial theorem to expand \((x+h)^3\): \[ (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \].

Step 4: 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: \[ f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} \].

Step 5: 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: \[ f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2) \].

Step 6: Evaluate the Limit

Evaluate the limit as \(h \to 0\): \[ f'(x) = 3x^2 + 3x(0) + (0)^2 \]. Simplifying gives: \[ f'(x) = 3x^2 \].

Key Concepts

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.

When calculating derivatives, different rules (such as the power rule, product rule, etc.) can be applied depending on the function's form. However, these rules are derived from the fundamental definition of the derivative itself, which is often using limits.

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.

Through this definition, all derivative rules originate, emphasizing its importance in calculus.

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 \( h \to 0 \).

Using the binomial theorem, functions like \( (x+h)^3 \) can be expanded systematically, simplifying the application of the limit definition of derivatives.