Problem 9 - Calculus

Question

Find the derivatives of the functions \(9-18:\) \(\cosh (3 x+1)\)

Step-by-Step Solution

Answer

The derivative is \( 3 \sinh(3x + 1) \).

Step 1: Review the Function

The function we need to differentiate is \( \cosh(3x + 1) \). Here, the hyperbolic cosine function, \( \cosh(u) \), has the inner function \( u = 3x + 1 \).

Step 2: Recall the Derivative Rule

The derivative of \( \cosh(u) \) with respect to \( u \) is \( \sinh(u) \). Thus, if we can express the function in the form \( \cosh(u) \), we can use this derivative rule.

Step 3: Differentiate the Inner Function

We need to first differentiate the inner function \( u = 3x + 1 \) with respect to \( x \). The derivative of \( u = 3x + 1 \) is \( u' = 3 \).

Step 4: Apply the Chain Rule

Use the chain rule to differentiate \( \cosh(3x + 1) \). According to the chain rule, the derivative \( \frac{d}{dx} \cosh(3x + 1) \) is \( \sinh(3x + 1) \times (3) \), which simplifies to \( 3 \sinh(3x + 1) \).

Step 5: Write the Final Derivative

Thus, the derivative of \( \cosh(3x + 1) \) with respect to \( x \) is \( 3 \sinh(3x + 1) \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are a set of mathematical functions that resemble trigonometric functions but are based on hyperbolas instead of circles. In calculus, we often deal with hyperbolic functions such as hyperbolic cosine (\(\cosh(x)\)) and hyperbolic sine (\(\sinh(x)\)). These functions have properties similar to their trigonometric counterparts but differ in terms of definitions and uses.

The hyperbolic cosine function is given by the formula:\[\cosh(x) = \frac{e^x + e^{-x}}{2}\]This function, like the regular cosine function, is even, meaning it is symmetric about the y-axis. On the other hand, the hyperbolic sine function is given as:\[\sinh(x) = \frac{e^x - e^{-x}}{2}\]The function \(\sinh(x)\) is odd, exhibiting symmetry about the origin.

These functions are central to many fields, including engineering and physics, particularly in dealing with exponential growth models and solving differential equations. Understanding their formulation and properties is crucial for mastery in calculus and beyond.

Derivative Rules

To differentiate functions, we use various rules that simplify the process. One such rule is the derivative of hyperbolic functions. For the hyperbolic cosine function \(\cosh(u)\), the rule is:

• The derivative of \(\cosh(u)\) with respect to \(u\) is \(\sinh(u)\)

This rule is similar to the derivative of the regular cosine, which is the regular sine, though specific to hyperbolic functions.

Derivative rules help in finding slopes of tangent lines and rates of change. They are fundamental tools in calculus for solving more complex problems. When applying these rules, attention to other functions such as polynomials or logarithms can be crucial, allowing seamless manipulation of various kinds of mathematical functions.

Learning these rules requires practice and understanding of how these functions behave. Familiarity with these rules can simplify the process of solving calculus problems efficiently.

Chain Rule

The chain rule is a vital derivative rule in calculus that allows us to differentiate composite functions. A composite function is a function that contains another function inside it. The chain rule states:

• If \(y = f(g(x))\), then the derivative \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\)

In simple terms, you multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function itself.

For example, to differentiate \(\cosh(3x + 1)\), first find the derivative of the outer function \(\cosh(u)\), which is \(\sinh(u)\). Then, multiply that by the derivative of the inner function \(3x + 1\), which is \(3\)
. This gives \(3 \sinh(3x + 1)\), illustrating how the chain rule simplifies differentiation of complex functions.

The chain rule is essential for correctly differentiating functions where one variable depends on another through intermediate functions. Mastery of this rule means quicker and more accurate problem-solving in calculus.