Problem 57 - Thomas Calculus

Question

In Exercises \(53-58,\) find the value of \((f \circ g)^{\prime}\) at the given value of \(x\) . $$ f(u)=\frac{2 u}{u^{2}+1}, \quad u=g(x)=10 x^{2}+x+1, \quad x=0 $$

Step-by-Step Solution

Answer

The value of \((f \circ g)^{\prime}(0)\) is 0.

Step 1: Determine function composition

The composite function \((f \circ g)(x)\) means \(f(g(x))\). First, substitute \(g(x) = 10x^2 + x + 1\) into \(f(u)\), which gives: \[ f(g(x)) = \frac{2(10x^2 + x + 1)}{(10x^2 + x + 1)^2 + 1}. \]

Step 2: Differentiate the composite function

To find \((f \circ g)^{\prime}(x)\), we utilize the chain rule, which states: \((f \circ g)^{\prime}(x) = f^{\prime}(g(x)) \cdot g^{\prime}(x)\). We proceed to find \(f^{\prime}(u)\) and \(g^{\prime}(x)\).

Step 3: Differentiate the function \(f(u)\)

Using the quotient rule for differentiation, \( f(u) = \frac{2u}{u^2 + 1}\), the derivative is given by \( f^{\prime}(u) = \frac{(2)(u^2 + 1) - (2u)(2u)}{(u^2 + 1)^2} = \frac{2 - 2u^2}{(u^2 + 1)^2}.\)

Step 4: Differentiate \(g(x)\)

The function \(g(x) = 10x^2 + x + 1\) is a polynomial, and its derivative is \( g^{\prime}(x) = 20x + 1.\)

Step 5: Evaluate at \(x = 0\)

First, find \( g(0)\), which is 1, so we substitute \(u = 1\) in \(f^{\prime}(u)\). Thus, \( f^{\prime}(1) = \frac{2 - 2(1)^2}{(1^2 + 1)^2} = \frac{0}{4} = 0\). Also, \( g^{\prime}(0) = 1\). Therefore, \( (f \circ g)^{\prime}(0) = 0 \cdot 1 = 0.\)

Key Concepts

Function Composition

Function composition involves combining two mathematical functions. One function acts upon the result of another. When dealing with composite functions, we denote them as \((f \circ g)(x) = f(g(x))\). This format indicates that we evaluate the inner function, in this case \(g(x)\), and use its result as the input to the outer function \(f\).

In the given exercise, we have \(f(u) = \frac{2u}{u^2+1}\) and \(g(x) = 10x^2 + x + 1\). To compose \((f \circ g)(x)\), substitute \(g(x)\) into \(f(u)\). This substitution step results in:

• First, compute \(g(x) = 10x^2 + x + 1\).
• Then, plug \(g(x)\) into \(f\), giving: \[f(g(x)) = \frac{2(10x^2 + x + 1)}{(10x^2 + x + 1)^2 + 1}.\]

The composition opens a pathway to determining how changes in \(x\) affect \(f\) through \(g\), paving the way for derivative analysis.

Quotient Rule

The quotient rule is vital in differentiating functions that are ratios of two expressions. It applies when differentiating a function \(f(u) = \frac{h(u)}{k(u)}\). The rule provides a structured formula: \[f'(u) = \frac{h'(u)k(u) - h(u)k'(u)}{(k(u))^2}\]. This formula helps unravel the differentiation of complex fractions.

In our problem, the function \(f(u) = \frac{2u}{u^2 + 1}\) becomes the focus. To differentiate this using the quotient rule:

• Identify top as \(h(u) = 2u\) and bottom as \(k(u) = u^2 + 1\).
• Compute the derivatives: \(h'(u) = 2\) and \(k'(u) = 2u\).
• Apply the quotient rule: \[f'(u) = \frac{2(u^2 + 1) - 2u(2u)}{(u^2 + 1)^2} = \frac{2 - 2u^2}{(u^2 + 1)^2}.\]

By following these steps, we derive an understanding of how the numerator and denominator contribute to the overall rate of change.

Derivative Evaluation

Evaluating a derivative at a specific point requires substituting the given value into the derivative function and calculating the resultant value. This process enables us to understand the function's behavior and rate of change at that particular point.

In the exercise, after using the chain rule, we find \((f \circ g)'(x) = f'(g(x)) \cdot g'(x)\). Evaluating this at \(x = 0\) involves the following steps to ensure precision:

• First, find \(g(0)\); here, it results in 1 (as \(g(0) = 10(0)^2 + 0 + 1 = 1\)).
• Substituting this into \(f'(u)\), we recognize \[f'(1) = \frac{2 - 2(1)^2}{(1^2 + 1)^2} = \frac{0}{4} = 0.\]
• Next, compute \(g'(0) = 20(0) + 1 = 1\).
• The final multiplication \((f \circ g)'(0) = 0 \cdot 1 = 0\).

Each detail of this evaluation precisely measures the function's change rate and confirms the solution for the given point.