Problem 38 - Brief Applied Calculus

Question

Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ \begin{array}{l} \underline{\phantom{xxx}} f(x)=x^{4} \\ \text { [Hint: Use } \quad(x+h)^{4}= \\ x^{4}+4 x^{3} h+6 x^{2} h^{2}+4 x h^{3}+h^{4} \end{array} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = x^4 \) is \( f^{\prime}(x) = 4x^3 \).

Step 1: State the Definition of the Derivative

The derivative of a function \( f(x) \) is defined as \[f^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}.\] In this case, we need to find \( f^{\prime}(x) \) for the function \( f(x) = x^4 \).

Step 2: Substitute f(x+h) and f(x) into the Derivative Formula

Using the given hint, replace \( f(x+h) \) with the expanded form: \( f(x+h) = (x+h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \).Substitute into the derivative definition:\[f^{\prime}(x) = \lim_{{h \to 0}} \frac{x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 - x^4}{h}.\]

Step 3: Simplify the Expression

Cancel \( x^4 \) from the numerator: \( \lim_{{h \to 0}} \frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h} \).Factor out \( h \) from the entire expression:\[ f^{\prime}(x) = \lim_{{h \to 0}} \frac{h(4x^3 + 6x^2h + 4xh^2 + h^3)}{h}. \]Cancel the \( h \):\[ f^{\prime}(x) = \lim_{{h \to 0}} (4x^3 + 6x^2h + 4xh^2 + h^3). \]

Step 4: Evaluate the Limit as h Approaches 0

Now, evaluate the limit as \( h \) approaches 0:\[f^{\prime}(x) = 4x^3 + 6x^2(0) + 4x(0)^2 + (0)^3.\]This simplifies to:\[ f^{\prime}(x) = 4x^3.\] Finally, the derivative of \( f(x) = x^4 \) is \( 4x^3 \).

Key Concepts

Limit Evaluation

To find the derivative of a function at a certain point, we use a powerful tool known as limit evaluation. This concept helps us to calculate the rate at which a function changes as we move towards an infinitesimally small point. The derivative, in essence, is the slope of the tangent line to the curve at a given point.

The formula for the derivative using limits is \[f^{\prime}(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}.\] Here, \(f(x+h)\) represents the function value at a slightly shifted point \(x+h\), and \(f(x)\) is the function value at \(x\). The difference \(f(x+h) - f(x)\) gives us a change in the function value with respect to \(x\). Dividing this change by \(h\), the shift, approximates the slope of the function. As \(h\) approaches zero, this approximation becomes exact.

In practical terms, this means pulling apart the function into its core components using known rules, like binomials, and carrying out algebraic simplifications until you can evaluate the limit.

Polynomial Differentiation

Polynomial differentiation is a specific application of the derivative concept, focusing on polynomials which are expressions like \(x^4\), where each term includes a variable raised to a power. When differentiating a polynomial function, each term is treated separately using a straightforward rule:

• The derivative of \(x^n\) is \(nx^{n-1}\).

Essentially, this rule tells us to multiply the original term by the power of \(x\) and then decrease the power by one.

To illustrate, the original function here is \(f(x) = x^4\). Applying the rule, the derivative becomes \(4x^3\). This result stems directly from applying the power rule to each term, confirming that polynomial differentiation is consistent and predictable.

Understanding this method is crucial for differentiating not only simple expressions but also more complex multi-term polynomials, where this process must be repeated for each term.

Expansion of Binomials

The expansion of binomials is a technique often used when evaluating derivatives through limit definitions. A binomial expression is one that includes two terms, like \((x + h)^n\). To differentiate polynomial functions effectively using limits, the binomial must be expanded.

A useful tool for expansion is the Binomial Theorem, which states:
\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^{k}.\] This theorem allows us to expand \((x+h)^4\) into a sum of terms: \(x^4 + 4x^3 h + 6x^2 h^2 + 4 x h^3 + h^4\). Each term corresponds to a combination of powers of \(x\) and \(h\) multiplied by coefficients.

In practice, this means substituting \(f(x+h)\) with its expanded form using binomial expansion. It simplifies subsequent algebraic manipulation required in differentiating by limits. This step is pivotal for finding derivatives of higher-degree polynomials without direct application of the power rule.