Problem 49 - Calculus Single Variable

Question

Differentiate the given expression with respect to \(x\). $$ \arctan (\sinh (x)) $$

Step-by-Step Solution

Answer

The derivative is \(\text{sech}(x)\).

Step 1: Identify the Chain Rule

The function to differentiate is a composition of two functions: the outer function is \(\arctan(u)\) and the inner function is \(u = \sinh(x)\). To differentiate this, we need to apply the chain rule.

Step 2: Differentiate the Outer Function

We differentiate the outer function \(\arctan(u)\) with respect to \(u\). The derivative is \(\frac{d}{du} \arctan(u) = \frac{1}{1+u^2}\).

Step 3: Differentiate the Inner Function

Now, differentiate the inner function \(u = \sinh(x)\) with respect to \(x\). The derivative is \(\frac{d}{dx} \sinh(x) = \cosh(x)\).

Step 4: Apply the Chain Rule

According to the chain rule, the derivative of a composition \(f(g(x))\) is given by \(f'(g(x)) \cdot g'(x)\). Apply this to our function: \(\frac{d}{dx} \arctan(\sinh(x)) = \frac{1}{1+\sinh^2(x)} \cdot \cosh(x)\).

Step 5: Simplify the Expression

Use the identity \(1 + \sinh^2(x) = \cosh^2(x)\) to simplify the expression. Thus, \(\frac{1}{1+\sinh^2(x)} = \frac{1}{\cosh^2(x)}\). This gives us the simplified derivative: \(\frac{d}{dx} \arctan(\sinh(x)) = \frac{\cosh(x)}{\cosh^2(x)} = \frac{1}{\cosh(x)}\). This is further simplified to \(\text{sech}(x)\).

Key Concepts

Derivative of inverse trigonometric functions

Inverse trigonometric functions, such as \(\arctan(\cdot)\), are crucial in mathematics for finding angles when given a ratio. Differentiating these functions requires special attention since each has a unique derivative rule. For instance, the derivative of \(\arctan(u)\) with respect to \(u\) is given by \(\frac{1}{1+u^2}\). This derivative rule is significant when differentiating expressions involving inverse trigonometric functions, especially when they are composed with other functions as seen in chain rule applications.Inverse trigonometric differentiation is not just a formula to memorize—it involves understanding how changes in one variable affect an angle in a right triangle. This means more practice will naturally build familiarity, making these differentiations more intuitive.

Hyperbolic functions

Hyperbolic functions like \(\sinh(x)\) and \(\cosh(x)\) resemble trigonometric functions, but their bases are exponential functions. They are defined as:

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

These functions play vital roles in various applications, including calculations involving hyperbolic angles and areas.When differentiating hyperbolic functions, each has a unique derivative rule much like their trigonometric counterparts. For instance, the derivative of \(\sinh(x)\) is \(\cosh(x)\). This derivative is particularly important when using these functions in the context of the chain rule, as seen when differentiating a composite function like \(\arctan(\sinh(x))\). Understanding these properties and identities, such as \(1 + \sinh^2(x) = \cosh^2(x)\), is crucial for simplifying the resulting expressions after differentiation.

Differentiation techniques

Differentiation techniques are the mathematical tools that allow us to find the rate at which one variable changes with respect to another. These techniques include methods like using the power rule, product rule, quotient rule, and particularly the chain rule, which is essential for dealing with composite functions.

The Chain Rule

The chain rule is a fundamental differentiation technique for composite functions \(f(g(x))\). It states that the derivative of \(f(g(x))\) is given by \(f'(g(x)) \cdot g'(x)\). In our exercise, this technique is employed to differentiate the composite function \(\arctan(\sinh(x))\). By differentiating the outer function \(\arctan(u)\) and multiplying it by the derivative of the inner function \(\sinh(x)\), we effectively find the rate of change of the entire expression.

Simplification

A crucial final step in differentiation often involves simplifying the expression, using identities or algebraic manipulation. For example, simplifying \(\frac{1}{1+ \sinh^2(x)}\cdot \cosh(x)\) to \(\text{sech}(x)\) can make the results more compact and easier to interpret. Understanding these differentiation techniques provides a more comprehensive ability to analyze and solve problems involving complex functions.