Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use \(f^{\prime}(x)=\lim _{t \rightarrow x}[f(t)-f(x)] /[t-x]\) to find \(f^{\prime}(x)\) (see Example 5). $$ f(x)=x^{3}+5 x $$

Step-by-Step Solution

Answer

The derivative \( f^{\prime}(x) \) is \( 3x^2 + 5 \).

Step 1: Substitute the Function into Derivative Formula

To find the derivative of the function \( f(x) = x^3 + 5x \), we use the formula for the derivative: \[ f^{\prime}(x) = \lim_{t \rightarrow x} \frac{f(t) - f(x)}{t-x} \]. Substitute \( f(t) \) as \( t^3 + 5t \) and \( f(x) \) as \( x^3 + 5x \).

Step 2: Compute the Difference

Calculate the difference \( f(t) - f(x) \):\[ f(t) - f(x) = (t^3 + 5t) - (x^3 + 5x) = t^3 - x^3 + 5t - 5x.\]

Step 3: Simplify the Expression

Factor the expression \( t^3 - x^3 + 5t - 5x \):\[ t^3 - x^3 = (t-x)(t^2 + tx + x^2),\]so,\[ f(t) - f(x) = (t-x)(t^2 + tx + x^2) + 5(t-x).\]Factor out \( t - x \):\[ f(t) - f(x) = (t-x)(t^2 + tx + x^2 + 5).\]

Step 4: Limit Calculation

Plug the expression back into the derivative formula:\[ f^{\prime}(x) = \lim_{t \rightarrow x} \frac{(t-x)(t^2 + tx + x^2 + 5)}{t-x}.\]Cancel out \( t-x \) in the numerator and denominator:\[ f^{\prime}(x) = \lim_{t \rightarrow x} (t^2 + tx + x^2 + 5).\]Now, apply the limit as \( t \rightarrow x \):\[ f^{\prime}(x) = x^2 + x^2 + x^2 + 5 = 3x^2 + 5.\]

Step 5: Conclusion

After evaluating the limit, the derivative of the function \( f(x) = x^3 + 5x \) is:\[ f^{\prime}(x) = 3x^2 + 5. \]

Key Concepts

Limit definition of a derivative

The limit definition of a derivative is a fundamental concept in calculus. The idea behind it is to find the rate of change, or slope, of a function at a particular point. This is done by evaluating the limit of the difference quotient as the change in the independent variable approaches zero. For example, given the function \[ f(x) = x^3 + 5x \], we can use the formula:

• \[ f^{\prime}(x) = \lim_{t \rightarrow x} \frac{f(t) - f(x)}{t-x} \].

Here, \( t \) is a point close to \( x \), and as \( t \) approaches \( x \), the expression provides the derivative at \( x \). Although solving the limit definition may involve several algebraic steps, it provides a precise result that reflects the instantaneous rate of change of the function.
By substituting \( f(t) = t^3 + 5t \) into the formula, the function's exact behavior is examined at every point \( x \). This approach beautifully bridges the gap between algebra and calculus, facilitating deeper understanding of how functions behave and change.

Polynomial functions

Polynomial functions are expressions built from variables and coefficients, typically composed of sums of terms of the form \( ax^n \) where \( n \) is a non-negative integer and \( a \) is a coefficient. In our exercise, the function \( f(x) = x^3 + 5x \) is a polynomial function of degree 3, which is the highest power of the variable \( x \).
Polynomials have specific characteristics based on their degree:

• For example, linear polynomials (degree 1) form straight lines.
• Quadratic polynomials (degree 2) create parabolas.
• Cubic polynomials (degree 3), like the one in our exercise, create curves that can change direction twice.

Understanding these properties is essential, as they dictate the general shape and behavior of the polynomial's graph. In this context, we explore how the function changes and apply derivative techniques to analyze those changes. These derivatives then afford us insights into the slope and turning points of the polynomial graph.

Differentiation techniques

Differentiation is the process of finding the derivative, which tells us the rate at which a function is changing at any point. Various techniques exist to differentiate functions, such as the power rule, product rule, and chain rule. However, when using the limit definition of the derivative, techniques mainly involve algebraic manipulation.
In this exercise, differentiating the polynomial \( f(x) = x^3 + 5x \) meant simplifying expressions to reach the derivative \( f^{\prime}(x) = 3x^2 + 5 \). This involves steps like:

• Calculating \( f(t) - f(x) \) to identify changes in the function.
• Factoring expressions to simplify the difference quotient.
• Cancelling common factors to simplify the expression further.

Applying the limit requires finding where expressions simplify or cancel out, ultimately obtaining a form free of indeterminate expressions. With \( f^{\prime}(x) = 3x^2 + 5 \), we gain information about the polynomial's slope at any point \( x \), helping us understand peaks, valleys, and inflection points of the function's curve.