Question

\(f(x)=\tan x, \quad a=\pi\)

Short answer

The derivative of the function \(f(x)=\tan x\) at \(x=\pi\) is 1.

Step-by-step solution

Calculation of Derivative

The derivative of \(f(x)=\tan x\) is \(f'(x)=\sec^2 x\). This formula comes from the standard differentiation rules.

Substituting \(x=a=\pi\)

On substituting \(x=\pi\) in \(f'(x)=\sec^2 x\), we get \(f'(\pi)=\sec^2 \pi\).

Evaluation of \(\sec^2 \pi\)

Here, \(\sec \pi = -1\) as secant function is reciprocal of cosine function and \(\cos \pi = -1\). Therefore, \(\sec^2 \pi = (-1)^2 = 1\).

Key concepts

Differentiation Rules

Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function is changing at any given point. It is the process of determining the derivative of a function. There are several rules for differentiation which simplifies this process for a variety of functions. Some of these rules include the power rule, product rule, quotient rule, and chain rule.

For trigonometric functions like the tangent function, specific rules apply. The derivative of a tangent function is not immediately obvious like a simple polynomial, but using the differentiation rules for trigonometric functions, we can determine that the derivative of \(\tan x\) is \(\sec^2 x\). This rule is crucial as it helps to solve a wide array of problems involving the tangent function in calculus.

Secant Function

The secant function, denoted as \(\sec x\), is one of the six basic trigonometric functions. It is defined as the reciprocal of the cosine function, essentially as \(\sec x = \frac{1}{\cos x}\). The secant function has its own unique graph, which illustrates that it possesses both positive and negative values and has undefined points where the cosine function equals zero.

The significance of the secant function in differentiation comes from the fact that it represents the slope of the tangent to the curve of the function \(y=\tan x\) at any point \(x\). Thus, understanding the secant function is critical for grasping the concepts involved in finding derivatives of trigonometric functions.

Reciprocal Trigonometric Functions

In trigonometry, the reciprocal trigonometric functions are those that invert the primary trigonometric functions, namely sine, cosine, and tangent. These reciprocals are the cosecant (\(\csc x = \frac{1}{\sin x}\)), secant (\(\sec x = \frac{1}{\cos x}\)), and cotangent (\(\cot x = \frac{1}{\tan x}\)) functions.

Understanding these reciprocal relationships is particularly useful in calculus when evaluating the derivatives of trigonometric functions. Their properties and the way they behave around important angles (like \(0\), \(\frac{\pi}{2}\), and \(\pi\)) are essential to correctly solving differentiation problems that involve these functions. For instance, the derivative of the tangent function involves the secant squared function, which showcases the importance of reciprocal functions in differentiation.

Evaluating Trigonometric Functions

Evaluating trigonometric functions involves finding the value of these functions at specific angles. Trigonometric functions take an angle as an input and produce a ratio as output. Understanding the values of trig functions at key angles is essential in calculus and trigonometry.

For example, knowing that \(\cos \pi = -1\) helps you quickly find that \(\sec \pi = -1\) since they are reciprocals of each other. This knowledge is applied in evaluating the derivative of the tangent function at \(\pi\), as seen in the solution for the derivative of \(\tan x\). Errors in evaluating these functions at particular angles can lead to mistakes in further calculations, making this skill critical for students to master.