Problem 9 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Identify the errors in the following derivative computations. \(\begin{array}{l}\text { a. }\left[\left(t^{4}+t^{-1}\right)^{7}\right]^{\prime} \\ & 7\left(t^{4}+t^{-1}\right)^{6}\left[t^{4}+t^{-1}\right]^{\prime} \\ & 7\left(t^{4}+t^{-1}\right)^{6}\left[t^{4}\right]^{\prime}+\left[t^{-1}\right]^{\prime} \\\ & 7\left(t^{4}+t^{-1}\right)^{6} 4 t^{3}+(-1) t^{-2} \\ & 28 t^{3}\left(t^{4}+t^{-1}\right)^{6}-t^{-2}\end{array}\) \(\begin{array}{l}\text { b. }\left[5 t^{7}+7 t^{-5}\right]^{\prime} \\\ {\left[5 t^{7}\right]^{\prime}+\left[7 t^{-5}\right]^{\prime}} \\ & 5\left[t^{7}\right]^{\prime}+7\left[t^{-5}\right]^{\prime} \\ & 5 \times 7 t^{6}+7 \times(-5) t^{-4} \\ & 35\left(t^{6}-t^{-4}\right)\end{array}\) c. \(\left[10 t^{8}+8 e^{5 t}\right]^{\prime}\) \(\left[10 t^{8}\right]^{\prime}+\left[8 e^{5 t}\right]^{\prime}\) \(10\left[t^{8}\right]^{\prime}+8\left[e^{5 t}\right]^{\prime}\) \(10 \times 8 t^{7}+8 \times 5 e^{4 t}\) \(40\left(2 t^{7}+e^{4 t}\right)\)

Step-by-Step Solution

Answer

Part a misapplies factors; Part b misuses parentheses; Part c alters components' multiplicity incorrectly.

Step 1: Review the Original Derivative Computation for Part a

In the expression provided, \(rac{d}{dt}[(t^4 + t^{-1})^7]\), the chain rule is used as the first step: \([7(t^4 + t^{-1})^6] \cdot [(t^4 + t^{-1})'] \). This part is correct.

Step 2: Identify the Error in Part a Differentiation

The differentiation inside should produce a product like \((t^4)' + (t^{-1})' = 4t^{3} + (-1)t^{-2}\), which was correctly derived. However, the mistake occurs in the application: the last expression reads erroneously as \(28t^3 (t^4 + t^{-1})^6 - t^{-2}\). The correct representation should maintain the combined factor: \([4t^3 - t^{-2}] \cdot 7(t^4 + t^{-1})^6 \).

Step 3: Review the Original Derivative Computation for Part b

The differentiation of \([5t^{7} + 7t^{-5}]'\) should clarify each term separately: \([5t^7]' + [7t^{-5}]'\). Each of these differentiations is correctly noted as \(35t^6 - 35t^{-4}\). The mistake occurs in aggregation: it writes \(35(t^6 - t^{-4})\) instead of \(35t^6 + (-35t^{-4})\).

Step 4: Identify the Error in Part b Aggregation

Ensure the negative sign is managed correctly when aggregating terms of differing polarity; the final notation has an extraneous parenthesis grouping which implies factoring that isn’t correct for the derivative representation.

Step 5: Review the Original Derivative Computation for Part c

Here, the expression \([(10t^8)' + (8e^{5t})']\) again involves taking derivatives separately as a sum: \(80t^7 + 40e^{5t}\). The mistake occurs in simplifying multipliers, falsely transposing the exponential factor: \(40(2t^7 + e^{4t})\) is incorrect due to wrong multiplier application.

Step 6: Correct the Derivative for Part c

Correct this by explicitly completing: the differentiation should aggregate to \(80t^7 + 40e^{5t}\), directly using the chain rule for the known factor exponential derivatives, instead of misapplying multipliers subsequently. The simplification is wrong due to a cardinal multiplier alteration and wrong component.

Key Concepts

Chain Rule

The Chain Rule is a powerful tool in calculus used to differentiate complex compositions of functions. It allows you to take the derivative of a nested function, where one function is nested inside another. To use the Chain Rule, you first identify the outer function and the inner function. Once these functions are determined, differentiate the outer function while keeping the inner function intact, then multiply by the derivative of the inner function. This process is incredibly important because it ensures the accurate differentiation of composite functions.

Here’s a generalized formula for the Chain Rule: if you have a composite function, say \( f(g(x)) \), the derivative is \( f'(g(x))g'(x) \). Start by differentiating the outer function \( f \), leaving \( g(x) \) unchanged, and finally multiply by the derivative of \( g(x) \). Using the Chain Rule correctly is crucial for solving many calculus problems accurately, and missing a component or not correctly applying it can lead to significant errors. This means you need to be diligent in identifying which part of the equation each component belongs to and carrying out each differentiation step precisely.

Differentiation Mistakes

Errors frequently occur when students fail to apply derivative rules correctly or overlook crucial components when simplifying their steps. Missed or incorrect differentiation application can lead to incorrect answers and misunderstandings of calculus concepts.

Common errors include:

• Omitting necessary components of composite functions in the Chain Rule, like the derivative of the inner function.
• Incorrectly simplifying expressions by miscalculating derivative components or forgetting to multiply terms.
• Mishandling signs, especially when combining terms, which can result in completely misaligned solutions.

To avoid these differentiation mishaps, always double-check each step in your work. Confirm that each derivative has been applied correctly, taking care to isolate any negative or fractional exponents accurately. By consistently reviewing work for these common errors, students can drastically reduce mistakes and see great improvements in their calculus problem-solving abilities.

Calculus Problem Solving

Effective calculus problem solving entails understanding the rules, methodologies, and processes for each specific type of calculus problem. When faced with a differentiation problem, such as the ones given in the original exercise, it is vital to clearly follow:

• The identification of differentiation rules needed, such as the Chain Rule, Product Rule, or simple power rule.
• The proper application of derivative rules to each part of the function.
• The correct aggregation and simplification of terms post-differentiation.

Analyzing each term separately, and ensuring all steps are logical and sequential, establishes correct solutions. Readability of steps and consistency in methods are key. Always double-check your final expression for potential errors in sign or original value misplacement. Utilize these problem-solving skills to enhance accuracy and efficiency in reaching the correct solutions.