Problem 36 - Brief Applied Calculus

Question

Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ \begin{array}{l} \underline{\phantom{xxx}} f(x)=(x+a)^{2}\\\ \left(a \text { is a constant.) [Hint: First expand }(x+a)^{2}\right. \text { .] } \end{array} $$

Step-by-Step Solution

Answer

The derivative is \( f^{\prime}(x) = 2x + 2a \).

Step 1: Expand the Function

The given function is \( f(x) = (x + a)^2 \). First, expand this expression using the formula \((x + a)^2 = x^2 + 2ax + a^2\). Thus, \( f(x) = x^2 + 2ax + a^2 \).

Step 2: Understand the Definition of the Derivative

The derivative \( f^{\prime}(x) \) using the definition is given by:\[f^{\prime}(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]We will apply this formula to find \( f^{\prime}(x) \).

Step 3: Calculate \( f(x + h) \)

Substitute \( x + h \) into the expanded function:\[f(x + h) = ((x + h) + a)^2 = (x + h + a)^2 = (x + a + h)^2\]Expand this to: \((x + a + h)^2 = (x + a)^2 + 2(x + a)h + h^2 = x^2 + 2ax + a^2 + 2xh + 2ah + h^2\).

Step 4: Substitute in the Derivative Formula

Now substitute \( f(x + h) \) and \( f(x) \) into the derivative formula:\[f^{\prime}(x) = \lim_{h \to 0} \frac{(x^2 + 2ax + a^2 + 2xh + 2ah + h^2) - (x^2 + 2ax + a^2)}{h}\]This simplifies to:\[f^{\prime}(x) = \lim_{h \to 0} \frac{2xh + 2ah + h^2}{h}\]

Step 5: Simplify the Expression

Factor out \(h\) from each term in the numerator:\[f^{\prime}(x) = \lim_{h \to 0} \frac{h(2x + 2a + h)}{h}\]Cancel \(h\) in the numerator and the denominator:\[f^{\prime}(x) = \lim_{h \to 0} (2x + 2a + h)\]

Step 6: Evaluate the Limit

Finally, substitute \( h = 0 \) into the simplified expression:\[f^{\prime}(x) = 2x + 2a\]Therefore, the derivative of \( f(x) = (x + a)^2 \) is \( f^{\prime}(x) = 2x + 2a \).

Key Concepts

Expansion of Algebraic Expressions

Understanding algebraic expressions is key to mastering calculus. When we expand an expression like \((x + a)^2\), we're essentially unpacking it into a more workable form. The expression \((x + a)^2\) when expanded, follows the rule for squaring a binomial: \[(x + a)^2 = x^2 + 2ax + a^2\].
This expansion process:

• Allows us to transform complex expressions into simpler forms.
• Reveals each term that can be separately analyzed and simplified.
• Makes differentiation and integration more straightforward.

Remember, every time we square a binomial, we apply this pattern. This expansion simplifies calculations, making subsequent mathematical operations much easier.

Limiting Process in Calculus

The concept of limits is a fundamental aspect of calculus. This is central to defining derivatives, like in our example problem. A limit helps understand the behavior of a function as it approaches a specific point.
In the definition of a derivative, the limit is expressed as:

• \(f^{\prime}(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\)

By examining this limit, we observe the slope of the tangent line to the curve at a point \(x\). To apply the limit:

• Calculate \(f(x+h)\), the value of the function a small increment \(h\) away from \(x\).
• Substitute it into the limit formula.
• Simplify until \(h\) approaches zero.

This approach reveals the instantaneous rate of change of the function. Understanding limits is crucial to finding derivatives and understanding the fundamental concepts of calculus.

Finding Derivatives Step-by-Step

Finding derivatives using the definition step-by-step provides a solid understanding of how calculus works. Here's how it unfolds:
**1. Start with Function Expansion**: Begin by expanding the given function, as in our example with \((x + a)^2\).
**2. Calculate \(f(x + h)\)**: Substitute \((x + h)\) into the expanded equation to express the function's value just beyond \(x\).
**3. Plug Into the Derivative Formula**: Using the difference quotient formula \(\frac{f(x + h) - f(x)}{h}\), replace and rearrange.
**4. Simplify and Solve the Limit**: Cancel out common terms, typically those involving \(h\), and watch them vanish as \(h\) approaches zero.
**5. Derive the Final Expression**: Complete the process by removing \(h\) as it approaches zero, finalizing your derivative.
This systematic approach ensures clarity, helping you deduce the slope of a tangent line effectively. Each step correlates to foundational calculus principles, reinforcing key concepts and providing a deeper learning experience.