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Magazine Chapter 8: Problem 11
Exer. \(3-26:\) Find \(f^{\prime}(x)\) if \(f(x)\) is the given expression. $$ \frac{\operatorname{sech}\left(x^{2}\right)}{x^{2}+1} $$
Short Answer
Expert verified
\( f'(x) = \frac{-2x\operatorname{sech}(x^2)[(x^2 + 1)\operatorname{tanh}(x^2) + 1]}{(x^2 + 1)^2} \).
Step by step solution
01
Identify the Differentiation Rules
To differentiate the given function \( f(x) = \frac{\operatorname{sech}(x^2)}{x^2 + 1} \), we will need to use the quotient rule for differentiation and the chain rule. The quotient rule states that if \( f(x) = \frac{u(x)}{v(x)} \), then the derivative \( f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} \). The chain rule will be used to differentiate \( \operatorname{sech}(x^2) \) and \( (x^2 + 1) \).
02
Define \(u(x)\) and \(v(x)\)
Let \( u(x) = \operatorname{sech}(x^2) \) and \( v(x) = x^2 + 1 \). We need to find \( u'(x) \) and \( v'(x) \) to apply the quotient rule.
03
Differentiate \(u(x)\) Using the Chain Rule
First, differentiate \( u(x) = \operatorname{sech}(x^2) \). The derivative of \( \operatorname{sech}(x) \) is \( -\operatorname{sech}(x)\operatorname{tanh}(x) \). Applying the chain rule, we get:\[ u'(x) = -\operatorname{sech}(x^2)\operatorname{tanh}(x^2) \cdot 2x \].
04
Differentiate \(v(x)\) Directly
Differentiate \( v(x) = x^2 + 1 \). The derivative is straightforward:\[ v'(x) = 2x \].
05
Apply the Quotient Rule
Substitute \( u'(x) \), \( v(x) \), \( v'(x) \), and \( u(x) \) back into the quotient rule formula:\[ f'(x) = \frac{(x^2 + 1) \cdot \left(-2x \operatorname{sech}(x^2) \operatorname{tanh}(x^2)\right) - \operatorname{sech}(x^2) \cdot (2x)}{(x^2 + 1)^2} \].
06
Simplify the Expression
Simplify the resulting expression:\[ f'(x) = \frac{-2x(x^2 + 1) \operatorname{sech}(x^2) \operatorname{tanh}(x^2) - 2x \operatorname{sech}(x^2)}{(x^2 + 1)^2} \].Combine the terms in the numerator to reflect the common factors.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quotient Rule
Differentiation, the process of finding the derivative of a function, often involves various rules and techniques. For the function \( f(x) = \frac{\operatorname{sech}(x^2)}{x^2 + 1} \), the quotient rule plays a crucial role. The quotient rule is used when you have a ratio of two functions, namely \( u(x) \) and \( v(x) \). It is stated as follows:
This systematic approach ensures that complex expressions are handled with ease.
- If \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} \).
This systematic approach ensures that complex expressions are handled with ease.
Chain Rule
Sometimes, functions are nested within one another, like \( \operatorname{sech}(x^2) \), requiring a different differentiation method known as the chain rule. This rule helps us differentiate compositions of functions and states:
Then, multiply by the derivative of the inner function, \( x^2 \), which gives us \( 2x \).
As a result, the chain rule allows us to manage the complexity of differentiating multi-layered functions easily.
- If a function \( y = f(g(x)) \), then its derivative is \( y' = f'(g(x)) \cdot g'(x) \).
Then, multiply by the derivative of the inner function, \( x^2 \), which gives us \( 2x \).
As a result, the chain rule allows us to manage the complexity of differentiating multi-layered functions easily.
Sech Function
The hyperbolic secant function, denoted as \( \operatorname{sech}(x) \), is the reciprocal of the hyperbolic cosine function \( \cosh(x) \). It is a valuable function in calculus and trigonometry, particularly when dealing with hyperbolic identities. Importantly, its derivative helps simplify expressions when applying the chain rule:
It ensures we can navigate through different steps smoothly, especially when these functions are part of a larger expression.
- The derivative of \( \operatorname{sech}(x) \) is \( -\operatorname{sech}(x)\operatorname{tanh}(x) \).
It ensures we can navigate through different steps smoothly, especially when these functions are part of a larger expression.
Calculus Problem
Calculus problems, much like the one we are tackling, often require blending different rules and methods. This specific problem involves finding the derivative for a function which is a quotient of \( \operatorname{sech}(x^2) \) and \( x^2 + 1 \). The combined use of the quotient and chain rules lays down the foundational approach in this context. By defining the parts of a function properly:
Each step intertwines, showcasing how integral these calculus tools are in tackling complex equations.
- Identify \( u(x) \) and \( v(x) \) for the quotient rule.
- Apply the chain rule effectively to differentiate \( \operatorname{sech}(x^2) \).
Each step intertwines, showcasing how integral these calculus tools are in tackling complex equations.
