Problem 71 - University Calculus: Early Transcendentals

Question

Verify the formulas in Exercises by differentiation. $$\int(7 x-2)^{3} d x=\frac{(7 x-2)^{4}}{28}+C$$

Step-by-Step Solution

Answer

The formula is verified as \\(\int(7x-2)^3 dx = \frac{(7x-2)^4}{28}+C\\).

Step 1: Differentiate the Right Side

Start by differentiating the right-hand side of the equation \( \frac{(7x-2)^4}{28} + C \). According to the derivative rules, the constant \( C \) disappears, and the derivative of a constant multiple of a function is the constant times the derivative of the function. Let \( u = 7x - 2 \), making the expression \( \frac{1}{28} u^4 \).

Step 2: Apply the Chain Rule

Using the chain rule, the derivative of \( u^4 \) is \( 4u^3 \), and then multiply by the derivative of \( u \) which is 7. Hence, the derivative is \( \frac{1}{28} \times 4(7x-2)^3 \times 7 \).

Step 3: Simplify the Expression

Simplify the expression from Step 2: \( \frac{1}{28} \times 4(7x-2)^3 \times 7 = \frac{28(7x-2)^3}{28} \). The 28 in the numerator and the 28 in the denominator cancel each other out, leaving \( (7x-2)^3 \).

Step 4: Compare the Result with the Original Expression

Compare the derived expression \( (7x-2)^3 \) with the integrated function \( \int (7x-2)^3 \, dx \). Since they are identical, the antiderivative is verified.

Key Concepts

Chain Rule

The chain rule is a fundamental technique in differentiation used when dealing with composite functions. It allows us to differentiate compositions of two or more functions, such as in our exercise where a function is raised to a power and has a linear expression within it. Suppose you have a composite function like \(f(g(x))\). To differentiate it, you will begin by taking the derivative of the outer function \(f\), evaluated at the inner function \(g(x)\), which is \(f'(g(x))\). Then, multiply this by the derivative of the inner function \(g(x)\), i.e., \(g'(x)\). Thus, the chain rule formula is expressed as:

• \(\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)\)

In our exercise, the function \((7x-2)^3\) was part of a larger expression. Here, setting \(u = 7x - 2\), the outer function is \(u^3\) and the inner function is \(7x - 2\). According to the chain rule, the derivative of \(u^3\) is \(3u^2\), and multiplying by \(\frac{du}{dx}\) gives \(3(7x-2)^2 \times 7\). This process results in the correct application of the chain rule, producing the derivative \(21(7x-2)^2\).

Antiderivative Verification

To ensure the antiderivative calculation is correct, we differentiate it and see if we arrive at the original function. In this exercise, the task was to verify the antiderivative \(\int (7x-2)^3 \, dx = \frac{(7x-2)^4}{28} + C\). By differentiating the right side, we should get back to \((7x-2)^3\). Begin by recognizing any constants or terms that simplify when deriving:

• The constant \(C\) from integration disappears because it differentiates to zero.
• The constant \(\frac{1}{28}\) acts as a scalar multiplier.

Differentiate the expression \(\frac{1}{28}(7x-2)^4\) using the chain rule, as explained earlier. Upon simplification, you arrive at \((7x-2)^3\) which confirms the antiderivative. This verification is crucial in calculus, as it checks whether our integration yields the right result, reaffirming the equivalence of differentiation and integration.

Derivative Rules

Derivative rules form the backbone of calculus, providing the guidelines needed to differentiate functions systematically. Some of the basic rules include:

• **Constant Rule:** The derivative of a constant \(C\) is \(0\).
• **Power Rule:** The derivative of \(x^n\) (where \(n\) is any real number) is \(nx^{n-1}\).
• **Constant Multiple Rule:** The derivative of \(cf(x)\), where \(c\) is a constant, is \(c \cdot f'(x)\).

In the provided solution, these rules play a crucial role. The constant \(\frac{1}{28}\) from \(\frac{(7x-2)^4}{28}\) uses the constant multiple rule, allowing it to be factored out and carried throughout the differentiating process. When differentiating terms like \((7x-2)^4\), the power rule is employed, first applying a simple form of change to the exponent and then multiplying by the original power. Using these derivative rules allows for a structured approach to solving calculus problems efficiently and accurately. Understanding them is essential for verifying complex derivatives as seen in this exercise.