Problem 32 - Calculus of a Single Variable

Question

In Exercises 23–32, find the derivative of the function. $$ g(x)=\operatorname{sech}^{2} 3 x $$

Step-by-Step Solution

Answer

The derivative of the function \(g(x) = sech^{2}{3x}\) is \(-6 \cdot sech^{2}{3x} \cdot tanh{3x}\).

Step 1: Identify the Outer and Inner Functions

For function \(g(x) = sech^{2}{3x}\), the outer function is \(f(u) = u^{2}\) and the inner function is \(h(x) = sech{3x}\). The chain rule states that the derivative of the composition of the two functions is \(f'(g(x)) \times g'(x)\), where \(f'\) is the derivative of \(f(u)\) and \(g'\) is the derivative of \(g(x)\).

Step 2: Derivative of the Outer Function

The derivative of \(f(u)=u^{2}\) is \(f'(u)=2u\). Therefore, the derivative of \(f(h(x))= sech^{2}{3x}\) based on the chain rule is \(2 \cdot sech{3x}\).

Step 3: Derivative of the Inner Function

The inner function is \(h(x)=sech{3x}\). The derivative of \(sech{x}\) is \(-sech{x} \cdot tanh{x}\). Therefore, applying the chain rule, the derivative of \(sech{3x}\) is \(-3 \cdot sech{3x} \cdot tanh{3x}\).

Step 4: Apply Chain Rule

Applying the chain rule as mentioned in Step 1, the derivative of \(g(x) = sech^{2}{3x}\) is \(f'(h(x)) \times h'(x) = 2 \cdot sech{3x} \times -3 \cdot sech{3x} \cdot tanh{3x}\).

Step 5: Simplify the Result

Simplify the result to obtain the final derivative of \(g(x) = sech^{2}{3x}\) as \(-6 \cdot sech^{2}{3x} \cdot tanh{3x}\).

Key Concepts

Chain Rule

The Chain Rule is a fundamental technique in calculus used to differentiate composite functions. It allows us to find the derivative of complex functions by breaking them down into simpler parts. When a function is composed of two other functions, say \( f(g(x)) \), the Chain Rule states that the derivative is:
\[(f \circ g)'(x) = f'(g(x)) \cdot g'(x)\]This means you multiply the derivative of the outer function by the derivative of the inner function.
Applying the Chain Rule requires careful identification of these component functions, which is an essential skill in calculus.

Outer and Inner Functions

Identifying the outer and inner functions in a composite function is a crucial step in applying the Chain Rule. In general, a composite function can be seen as \( f(g(x)) \), where \( f \) is the outer function and \( g(x) \) is the inner function.

• Outer Function: This is what you apply first in the composition. For example, if the function is \( \operatorname{sech}^2 (3x) \), the outer function is \( u^2 \).
• Inner Function: This comes inside the outer function. For the same example, the inner function is \( \operatorname{sech}(3x) \).

Understanding this layering is key to correctly applying the Chain Rule to find the derivative of a function.
Each function plays an essential role, and careful parsing of them enables effective differentiation.

Hyperbolic Functions

Hyperbolic functions, such as hyperbolic secant (\( \operatorname{sech}(x) \)), are analogs of trigonometric functions that arise frequently in calculus. They have properties similar to trigonometric functions but apply to hyperbolas.

• \( \operatorname{sech}(x) = \frac{1}{\cosh(x)} \): An analogy to the secant function in trig.
• The derivative of \( \operatorname{sech}(x) \) is \( -\operatorname{sech}(x) \cdot \operatorname{tanh}(x) \).

Hyperbolic functions demonstrate unique derivatives, making them important components in calculus problems.
These derivatives need to be efficiently linked and simplified when used in composite functions.

Simplification of Derivatives

Simplifying derivatives is often the final step after applying the Chain Rule, bringing the expression into a more manageable form. After deriving a function, the raw derivative may appear complex, requiring simplification.
In our example, after deriving \( \operatorname{sech}^2 (3x) \), the expression \(-6 \cdot \operatorname{sech}^2 (3x) \cdot \operatorname{tanh}(3x)\) is obtained through multiplication of individual derivatives identified by the Chain Rule.

• Combine like terms or constants, if applicable.
• Utilize identities or factorizations to reduce expressions.

Simplification not only makes the derivative easier to interpret, but it also assists in further computations required in complex problem-solving scenarios.