91Ó°ÊÓ

Hyperbolic functions are analogs of the trigonometric functions but for a hyperbola instead of a circle. Two common hyperbolic functions are the hyperbolic sine, \(\text{sinh}(x)\), and the hyperbolic cosine, \(\text{cosh}(x)\).
Understanding how to differentiate these functions is essential for solving problems involving higher-order derivatives. To derive these functions:

• The derivative of \(\text{sinh}(x)\) is \(\text{cosh}(x)\).
• The derivative of \(\text{cosh}(x)\) is \(\text{sinh}(x)\).

These derivatives form a cyclic pattern, similar to the way sine and cosine derivatives alternate. Knowing these basic derivatives will be very useful when differentiating products of polynomials and hyperbolic functions.

The product rule is a fundamental tool in calculus used to differentiate products of two functions. Suppose you have two functions, \(u(x)\) and \(v(x)\). The product rule states that:
\[\frac{d}{dx} [u(x) \, v(x)] = u'(x) \, v(x) + u(x) \, v'(x)\]
This rule can be applied iteratively to find higher-order derivatives. For instance, to differentiate \(y = x^{2} \, \text{sinh}(x)\), we apply the product rule as follows:
\[\frac{d}{dx} [x^2 \, \text{sinh}(x)] = 2x \, \text{sinh}(x) + x^2 \, \text{cosh}(x)\]
Repeating such steps will reveal the general pattern which helps in determining higher-order derivatives. Similarly, for \(y = x^{3} \, \text{cosh}(x)\), the first derivative is found using:
\[\frac{d}{dx} [x^3 \, \text{cosh}(x)] = 3x^2 \, \text{cosh}(x) + x^3 \, \text{sinh}(x)\]

When solving problems involving higher-order derivatives, it is important to identify derivative patterns. By repeatedly applying the product rule, you will often notice a pattern. For example:
For \(y = x^{2} \, \text{sinh}(x)\), and identifying after multiple differentiations, we see:

• First derivative: \( 2x \, \text{sinh}(x) + x^2 \, \text{cosh}(x) \)
• Second derivative: \( 6x \, \text{cosh}(x) + x^2 \, \text{sinh}(x) \)

Similarly, for \(y = x^{3} \, \text{cosh}(x)\):

• First derivative: \( 3x^2 \, \text{cosh}(x) + x^3 \, \text{sinh}(x) \)
• Second derivative: \( 15x \, \text{sinh}(x) + x^3 \, \text{cosh}(x) \)

Recognizing these patterns helps in generalizing the formula for the \((2n)\)th derivative. The final patterns for these specific functions are:
For \(y = x^{2} \, \text{sinh}(x)\): \[ (2n)! x^{2} \, \text{sinh}(x) \]
For \(y = x^{3} \, \text{cosh}(x)\): \[ (2n)! x^{3} \, \text{cosh}(x) \]