What Is The Differentiation Of Tan X

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Differentiation is a fundamental concept in calculus, used to determine the rate of change of a function. One important function in trigonometry is tan(x), which appears frequently in mathematics, physics, and engineering.

In this topic, we will explore the differentiation of tan(x), understand the formula and derivation, and look at real-life applications of this derivative.

1. Understanding the Function tan(x)

The function tan(x), or tangent of x, is a trigonometric function defined as:

text{tan}(x) = frac{sin(x)}{cos(x)}

It is periodic, meaning it repeats its values at regular intervals, with a period of π. However, it is undefined at x = frac{pi}{2}, frac{3pi}{2}, frac{5pi}{2},…, where the denominator cos(x) = 0.

2. The Differentiation Formula of tan(x)

The derivative of tan(x) with respect to x is given by:

frac{d}{dx} [tan(x)] = sec^2(x)

This result is fundamental in calculus and is widely used in solving integration, physics problems, and engineering applications.

3. Derivation of the Differentiation of tan(x)

To derive the formula for d/dx [tan(x)], we use the quotient rule of differentiation. The quotient rule states:

frac{d}{dx} left(frac{f(x)}{g(x)}right) = frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2}

Since tan(x) = sin(x) / cos(x), let:

  • f(x) = sin(x)
  • g(x) = cos(x)

Now, differentiate sin(x) and cos(x):

  • f'(x) = cos(x)
  • g'(x) = -sin(x)

Applying the quotient rule:

frac{d}{dx} [tan(x)] = frac{cos(x) cdot cos(x) – sin(x) cdot (-sin(x))}{cos^2(x)}
= frac{cos^2(x) + sin^2(x)}{cos^2(x)}

Using the Pythagorean identity:

sin^2(x) + cos^2(x) = 1

We get:

frac{1}{cos^2(x)} = sec^2(x)

Thus,

frac{d}{dx} [tan(x)] = sec^2(x)

4. Applications of the Derivative of tan(x)

4.1 In Physics

  • Used in motion equations involving angles, such as projectile motion.
  • Helps in analyzing wave mechanics and oscillations.

4.2 In Engineering

  • Used in designing inclined planes and slopes.
  • Important in electrical engineering for signal analysis.

4.3 In Economics and Business

  • Used in optimization problems involving marginal costs and revenues.

5. Examples of Differentiating tan(x)

Example 1: Basic Differentiation

Find the derivative of f(x) = tan(x) + 3x.

Solution:

f'(x) = sec^2(x) + 3

Example 2: Chain Rule Application

Differentiate g(x) = tan(5x).

Solution: Using the chain rule,

g'(x) = 5sec^2(5x)

Example 3: Second Derivative of tan(x)

Find the second derivative of h(x) = tan(x).

Solution:

h'(x) = sec^2(x)
h”(x) = 2sec^2(x)tan(x)

6. Key Takeaways

  • The differentiation of tan(x) is sec²(x).
  • It is derived using the quotient rule.
  • It has applications in physics, engineering, and economics.
  • Advanced problems use the chain rule and second derivatives.

The derivative of tan(x) is an essential concept in calculus with many real-world applications. By understanding how it is derived and applied, students and professionals can use it to solve complex mathematical and scientific problems efficiently.