Problem 31 - Calculus of a Single Variable

Question

In Exercises 23–32, find the derivative of the function. $$f(t)=\arctan (\sinh t)$$

Step-by-Step Solution

Answer

The derivative of the function \(f(t) = \arctan (\sinh(t))\) is \(f'(t) = \cosh(t)/(1+\sinh^2(t))\).

Step 1: Identify the Inner and Outer Functions

First, identify the inner and outer functions. Here, the inner function is \(g(t) = \sinh(t)\) and the outer function is \(f(g) = \arctan(g)\).

Step 2: Differentiate the Outer Function with the Chain Rule

Differentiate the outer function \(f(g) = \arctan(g)\) using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Because the derivative of arctan(x) is \(1/(1+x^2)\), the derivative of \(f(g) = \arctan(g)\), treating g like a constant, is \(f'(g) = 1/(1+g^2)\).

Step 3: Differentiate the Inner Function

Differentiate the inner function \(g(t) = \sinh(t)\). The derivative of the hyperbolic sine function is the hyperbolic cosine function, so \(g'(t) = \cosh(t)\).

Step 4: Apply the Chain Rule

Now apply the chain rule by multiplying the derivatives of the outer and inner functions. Substitute \(g(t) = sinh(t)\) into \(f'(g)\) to find \(df/dt = 1/(1+sinh^2(t)) \cdot \cosh(t)\).

Key Concepts

Understanding Hyperbolic Functions

Hyperbolic functions are analogs of the common trigonometric functions, but they are based on hyperbolas instead of circles. They are widely used in mathematics for their unique properties and applications in various fields like complex analysis and geometry.

Here are some common hyperbolic functions you should know:

• Sinh: The hyperbolic sine function, denoted by \( \sinh(t) \), is defined as \( \frac{e^t - e^{-t}}{2} \). It is equivalent to a coordinate on a hyperbola, just as sine is related to the unit circle.
• Cosh: The hyperbolic cosine function, denoted by \( \cosh(t) \), is defined as \( \frac{e^t + e^{-t}}{2} \). Like hyperbolic sine, it describes another aspect of a hyperbola.
• These functions have some interesting relationships, such as \( \cosh^2(t) - \sinh^2(t) = 1 \).
• Due to their definitions, hyperbolic functions are related to exponential functions, which gives them distinct and smooth curves.

Understanding these functions is crucial for analysis and computation involving hyperbolic identities and transformations.

Basics of Differentiation

Differentiation is the process of finding a derivative, which measures how a function changes as its input changes. In simpler terms, it tells us the slope of the tangent line to the curve of a function at any given point.

Here are some key points about differentiation:

• Derivative of Elementary Functions: The derivative of basic functions like powers, exponentials, and trigonometric functions are the building blocks of differentiation.
• Rules of Differentiation: Essential rules include the power rule, product rule, quotient rule, and the chain rule. These help simplify finding derivatives of complex functions.
• In practice, to differentiate any function, you apply these rules according to the structure of the function.
• It's important for understanding rates of change, such as acceleration, within physics and other sciences.

Remember, the chain rule is particularly helpful for differentiating composite functions, which we'll explore next.

Composite Functions and the Chain Rule

Composite functions are functions formed by combining two or more functions, where the output of one function becomes the input of another. This nesting of functions is common in calculus, especially when differentiating using the chain rule.

Here's how the chain rule applies:

• Identify the outer function and the inner function. In our example, \( f(g) = \arctan(g) \) is the outer function, and \( g(t) = \sinh(t) \) is the inner function.
• Differentiate the outer function, treating the inner function as a constant: \( f'(g) = \frac{1}{1+g^2} \).
• Next, find the derivative of the inner function: \( g'(t) = \cosh(t) \).
• Combine these using the chain rule: multiply the derivative of the outer function by the derivative of the inner function, applying substitutions as necessary.

This method efficiently finds the derivative of compound functions and is essential for working with models involving multiple variables and functions.