Problem 26 - Calculus of a Single Variable

Question

In Exercises 23–32, find the derivative of the function. $$ f(x)=\tanh \left(4 x^{2}+3 x\right) $$

Step-by-Step Solution

Answer

The derivative of \(f(x)=\tanh \left(4 x^{2}+3 x\right)\) is \(\text{sech}^2(4x^{2} + 3x) * (8x +3)\)

Step 1: Identify the original function and its inner function

In this case, the function is \(\tanh x\), and the inner function is \(h(x)=4x^{2}+3x\)

Step 2: Derive the outer function

The derivative of \(\tanh x\), which can be written as \(\frac{d}{dx}(\tanh x)\), is \(\text{sech}^2 x\)

Step 3: Derive the inner function

The derivative of the inner function \(h(x)=4x^2 +3x\) is \(h'(x) = 8x +3\)

Step 4: Apply the chain rule

Now, apply the chain rule \(\frac{d}{dx}f(h(x)) = f'(h(x)) * h'(x)\). This gives: \(\text{sech}^2(4x^{2} + 3x) * (8x +3)\)

Key Concepts

Understanding the Chain Rule

The chain rule is a fundamental concept in calculus used when differentiating composite functions. A composite function is a function that contains another function within it. The chain rule allows us to differentiate such functions seamlessly by breaking them into simpler parts.

The basic idea is to differentiate the outer function first and then multiply it by the derivative of the inner function. In simple terms, if you have a function of the form \(f(g(x))\), the derivative \(f'(x)\) is \(f'(g(x)) \cdot g'(x)\).

• First, differentiate the outer function as if the inner function is a single variable.
• Next, multiply it by the derivative of the inner function.

This is exactly what was done in the original step-by-step solution: the derivative of \(\tanh(4x^2 + 3x)\) was found using the chain rule. Understand this method, and differentiating composite functions will become a breeze.

Hyperbolic Functions Explained

Hyperbolic functions are analogs of the trigonometric functions but for the hyperbola, just as the sine and cosine functions are for the circle. The hyperbolic function \(\tanh(x)\) is analogous to the tangent function but in hyperbolic geometry.

• \(\tanh(x)\) is defined as \(\frac{\sinh(x)}{\cosh(x)}\), where \(\sinh\) and \(\cosh\) are hyperbolic sine and cosine, respectively.
• These functions have properties and derivatives similar to their trigonometric counterparts, making them useful in various calculus problems.

Understanding hyperbolic functions and their purpose will help you see why their derivatives are structured as they are, and why they appear frequently in calculus problems related to hyperbolic geometry or differential equations.

What is the Sech Function?

The sech function, or hyperbolic secant, is crucial when working with hyperbolic functions like \(\tanh(x)\). It is defined as the reciprocal of the hyperbolic cosine function \(\cosh(x)\), meaning that \(\text{sech}(x) = \frac{1}{\cosh(x)}\).

When differentiating \(\tanh(x)\), the result involves \(\text{sech}^2(x)\), which signifies how the hyperbolic tangent's slope behaves. Recognizing this derivative's form is vital:

• It's the hyperbolic analogy to the square of the secant function in trigonometry.
• In calculus and advanced mathematics, it helps in solving differential equations involving hyperbolic functions.

Mastering the sech function broadens your understanding of hyperbolic identities and their applications in calculus differentiation.

The Basics of Calculus Differentiation

Differentiation is a core aspect of calculus and measures how a function changes as its input changes. It is used to determine the rate at which one quantity changes with respect to another and is essential for understanding the behavior of mathematical models.

In the context of the exercise provided, we used differentiation to find the derivative of a hyperbolic function. It's crucial to learn basic derivative rules and how they apply to various types of functions

• Simple polynomial, trigonometric, and exponential functions have well-known derivative rules.
• When dealing with more complex functions, tools like the chain rule become indispensable.

The foundation of calculus differentiation lies in understanding these rules and how to apply them, which ensures that you can tackle a wide array of problems effectively. Keeping these ideas in mind will help you differentiate not only standard functions but also composite ones efficiently.