Problem 40 - Calculus Single Variable

Question

Differentiate the given expression with respect to \(x\). $$ \cosh ^{2}(x) $$

Step-by-Step Solution

Answer

The derivative of \( \cosh^2(x) \) is \( \sinh(2x) \).

Step 1: Understand the Problem

We need to differentiate the expression \( \cosh^2(x) \) with respect to \( x \). \( \cosh(x) \) is the hyperbolic cosine function.

Step 2: Use the Chain Rule

To differentiate \( \cosh^2(x) \), we recognize it as a composite function: \( u(x) = \cosh(x) \) and \( f(u) = u^2 \). The chain rule states that the derivative of \( f(u(x)) \) is \( f'(u(x)) \cdot u'(x) \).

Step 3: Differentiate the Outer Function

The outer function is \( f(u) = u^2 \), whose derivative is \( f'(u) = 2u \). Therefore, \( f'(\cosh(x)) = 2\cosh(x) \).

Step 4: Differentiate the Inner Function

The inner function is \( u(x) = \cosh(x) \), whose derivative is \( u'(x) = \sinh(x) \), since the derivative of \( \cosh(x) \) is \( \sinh(x) \).

Step 5: Apply the Chain Rule

Using the chain rule, multiply the derivatives of the inner and outer functions: \( 2\cosh(x) \cdot \sinh(x) \).

Step 6: Simplify the Expression

The derivative is \( 2\cosh(x)\sinh(x) \), which is equivalent to \( \sinh(2x) \) based on the hyperbolic identity: \( \sinh(2x) = 2\sinh(x)\cosh(x) \).

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus, especially for differentiating composite functions. When we have a function inside another function, like in the case of \(\cosh^2(x)\), the chain rule helps us find the derivative effectively.
By definition, if we have a composite function \(f(g(x))\), the chain rule states:

• Differentiate the outer function with respect to the inner function.
• Differentiate the inner function with respect to \(x\).
• Multiply the two derivatives together.

In our example, the outer function is \(u^2\) with \(u = \cosh(x)\), and its derivative is \(2u\). The inner function is \(\cosh(x)\), and its derivative is \(\sinh(x)\). Finally, multiplying these results gives \(2\cosh(x)\sinh(x)\), illustrating how the chain rule works in practice.

Hyperbolic Functions

Hyperbolic functions, are analogues of trigonometric functions but for a hyperbola instead of a circle. The hyperbolic cosine, \(\cosh(x)\), and hyperbolic sine, \(\sinh(x)\), are the most common. They are defined using exponential functions:

\[\cosh(x) = \frac{e^x + e^{-x}}{2}\]
\[\sinh(x) = \frac{e^x - e^{-x}}{2}\]

These functions have unique properties that are useful in calculus. For instance, the derivative of \(\cosh(x)\) is \(\sinh(x)\), and vice versa, the derivative of \(\sinh(x)\) is \(\cosh(x)\).
This convenient relationship simplifies calculations when differentiating expressions involving hyperbolic functions, such as in this exercise. By understanding the properties of these functions, calculus problems become easier to solve.

Calculus Problem Solving

Solving calculus problems effectively requires a systematic approach. Here is a general strategy you can use, and it applies well to differentiating expressions like \(\cosh^2(x)\):

• Clearly understand the expression you need to differentiate.
• Identify it as a simple or composite function.
• If composite, recognize the inner and outer functions.
• Apply rules such as the chain rule, power rule, or others as needed.
• Simplify your final expression.

By following these steps, you develop a structured method of tackling problems and reduce the chances of mistakes. In our example, understanding each function's role, applying the correct derivative rules, and simplifying using hyperbolic identities result in precise solutions. This problem-solving skill speeds up calculation and enhances your ability to handle more complex calculus problems.