Problem 247 - Calculus Volume 1

Question

Use the information in the following table to find \(h^{\prime}(a)\) at the given value for \(a\). $$\begin{array}{|c|c|c|c|c|} \hline x & f(x) & f^{\prime}(x) & g(x) & g^{\prime}(x) \\ \hline 0 & 2 & 5 & 0 & 2 \\ \hline 1 & 1 & -2 & 3 & 0 \\ \hline 2 & 4 & 4 & 1 & -1 \\ \hline 3 & 3 & -3 & 2 & 3 \\ \hline \end{array}$$ $$ h(x)=\left(x^{4}+g(x)\right)^{-2} ; a=1 $$

Step-by-Step Solution

Answer

The derivative \( h'(1) \) is \(-\frac{1}{8}\).

Step 1: Understand the Problem

We need to find the derivative of the function \( h(x) = \left(x^4 + g(x)\right)^{-2} \) at \( x = a = 1 \). We have the function \( g(x) \) and its derivative \( g'(x) \) given in the table for various values of \( x \).

Step 2: Use the Chain Rule for Derivatives

The function \( h(x) = \left(x^4 + g(x)\right)^{-2} \) is a composite function. To find \( h'(x) \), we'll apply the chain rule: if \( h(x) = U^{-2} \), where \( U = x^4 + g(x) \), then \( h'(x) = -2U^{-3} \cdot U'(x) \).

Step 3: Find U and Its Derivative U'

Let \( U = x^4 + g(x) \). Then \( U'(x) = \frac{d}{dx}(x^4) + \frac{d}{dx}(g(x)) = 4x^3 + g'(x) \).

Step 4: Evaluate U and U' at x = 1

Substitute \( x = 1 \) into \( U \) and \( U' \):\( U = 1^4 + g(1) = 1 + 3 = 4 \).From the table, \( g'(1) = 0 \), so \( U'(1) = 4(1)^3 + 0 = 4 \).

Step 5: Substitute Values into the Derivative Formula

We now substitute the known values into the expression for \( h'(x) \):\[ h'(1) = -2(4)^{-3} \cdot 4 \].Calculate the expression:\[ h'(1) = -2 \cdot \frac{1}{64} \cdot 4 = -\frac{8}{64} = -\frac{1}{8} \].

Step 6: Conclusion: Find the Result

The derivative \( h'(x) \) evaluated at \( x = 1 \) is thus \(-\frac{1}{8}\).

Key Concepts

chain rule

The chain rule is a fundamental principle in calculus, particularly useful when dealing with composite functions. Composite functions are two or more functions combined, where the output of one function becomes the input of another.
The rule helps us find the derivative of such functions. Here’s how it works: if you have a function expressed as \( h(x) = (U(x))^n \), where \( U(x) \) is some function of \( x \), the chain rule states that the derivative of \( h(x) \) is \( h'(x) = n(U(x))^{n-1} \cdot U'(x) \).
For practical application:

• Identify the outer and inner functions in your expression.
• Differentiate the outer function, keeping the inner function unchanged.
• Multiply this result by the derivative of the inner function.

This method allowed us to solve for \( h'(x) \) by setting \( U = x^4 + g(x) \), differentiating, and eventually finding \( h'(1) = -\frac{1}{8} \). Understanding and mastering the chain rule is essential for tackling any calculus problem involving composite functions.

table of derivatives

A table of derivatives is a valuable tool for quick reference in calculus. It typically includes a list of basic derivatives for standard functions such as polynomials, exponentials, trigonometric, and logarithmic functions.
In the context of the exercise above, the table provides key values of functions and their derivatives at specific points.
Using the example given, the table tells us:

• \( f(x), g(x) \) for different values of \( x \)
• Their corresponding derivatives, \( f'(x), g'(x) \)

For instance, knowing that \( g'(1) = 0 \) directly from the table enables us to compute \( U'(1) = 4 \cdot (1)^3 + 0 = 4 \).
Accessing such tabular data can simplify derivative problems significantly, allowing you to skip the computation of basic derivatives and focus on more complex applications.

calculus problem-solving

Solving calculus problems often requires a combination of various concepts and techniques, like the chain rule and derivative tables.
Effective problem-solving in calculus follows a structured approach:

• Understand the question: Clearly identify the function or equations involved in the problem.
• Identify known values: Use given data, such as values in a table, to fill in parts of the equation.
• Apply relevant rules: Select the appropriate calculus rules, such as the chain rule for composite functions, to simplify and differentiate expressions.
• Substitute and solve: Input known values and calculate step by step to get the solution.

In the solved example, we calculated the derivative at a specific point by:

1. Understanding the necessity of the chain rule due to the function's composite nature.
2. Using a table to find relevant values of \( g(x) \) and \( g'(x) \) at \( x = 1 \).
3. Substituting these into our differentiated expression to find \( h'(x) \).

Mastering these problem-solving techniques in calculus not only aids in solving textbook exercises but also builds a strong foundation for more complex mathematical challenges.