Question

Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\). \(f(x)=\tan x\)

Short answer

Question: Given the function \(f(x) = \tan x\), determine the relationship between a small change in \(x\) (denoted as \(dx\)) and the corresponding change in \(y\) (denoted as \(dy\)). Answer: For the given function, the relationship between the small change in \(x\) (denoted as \(dx\)) and the change in \(y\) (denoted as \(dy\)) is \(dy = (\sec^2 x)dx\).

Step-by-step solution

Identify the given function

Let's first identify the given function \(f(x)=\tan x\) that we will need to analyze.

Compute the derivative of the function

Next, we need to find the derivative \(f'(x)\) of the given function \(f(x)\). This can be done using the chain rule. Recall that \(\frac{d}{dx}(\tan x) = \sec^2 x\). Therefore, our derivative is \(f'(x) = \sec^2 x\).

Determine the relationship between \(dy\) and \(dx\)

We are now ready to find the relationship between the small change in \(x\) (denoted as \(dx\)) and the corresponding change in \(y\) (denoted as \(dy\)) in the form \(dy = f'(x)dx\). Using the derivative found in Step 2, the relationship becomes: \(dy = (\sec^2 x)dx\). And that's the final answer. For the given function, the relationship between the small change in \(x\) (denoted as \(dx\)) and the change in \(y\) (denoted as \(dy\)) is \(dy = (\sec^2 x)dx\).

Key concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions. In trigonometry, it is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. For a given angle \( x \), the tangent is denoted as \( \tan x \). In terms of the unit circle, which is often used in calculus, \( \tan x \) can be thought of as the slope of the line that represents the angle in the circle.
Determinants:

• \( \tan x \) is periodic with a period of \( \pi \) radians.
• It's undefined for angles where cosine is zero (e.g., \( \pi/2, 3\pi/2, \, \dots \)).

Understanding the behavior and properties of the tangent function is crucial when working on problems involving derivatives. It provides a good practice of differentiating trigonometric functions and sets the foundation for more complex calculus applications.

Derivative of Trigonometric Functions

Calculating the derivative of trigonometric functions is an important skill in differential calculus. For the tangent function, the derivative is particularly interesting:
The Derivative Formula for Tangent:

• The derivative of \( \tan x \) with respect to \( x \) is \( \sec^2 x \).

This stem from the fundamental trigonometric identities, specifically how \( \tan x = \frac{\sin x}{\cos x} \) can be differentiated using quotient rule or directly remembered as \( \sec^2 x \). The term \( \sec x \) is the secant function, which is \( \frac{1}{\cos x} \).
Key points to remember include:

• The slopes of the tangent lines to the curves of trigonometric functions at any given point describe the rate of change at that point.
• The differentiation of \( \tan x \) resulting in \( \sec^2 x \) highlights the relation between tangent, cosine, and secant functions.

Recognizing these derivatives helps in understanding more complex functions involving trigonometric identities.

Chain Rule

The chain rule is a vital concept in calculus when dealing with composite functions. It helps find the derivative of a function composed of other functions. In simpler terms, if a function \( g(x) \) is inside a function \( f(x) \), the derivative of the composite function \( f(g(x)) \) is computed as \( f'(g(x)) \times g'(x) \).
In the context of the tangent derivative, while the direct application isn't immediately obvious, the chain rule is crucial when extending this concept to functions like \( \tan(u(x)) \) where \( u(x) \) is itself a function of \( x \).
For instance, if \( u(x) = 3x \), then \( y = \tan(3x) \) would require:

• First, find the derivative of \( \tan u = \sec^2 u \).
• Then multiply by the derivative of \( u(x) \), which is \( 3 \).

This approach allows for evaluating derivatives of numerous function combinations, making the chain rule indispensable for solving many calculus problems effectively.