Derivative of Tan x

We now understand the derivative of tan x. It is commonly represented as d/dx (tan x) or (tan x)', and its value is sec²x. The function tan x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Tangent Function: (tan(x) = sin(x)/cos(x)).

Quotient Rule: Rule for differentiating tan(x) (since it consists of sin(x)/cos(x)).

Secant Function: sec(x) = 1/cos(x).

The derivative of tan x can be denoted as d/dx (tan x) or (tan x)'. The formula we use to differentiate tan x is:

d/dx (tan x) = sec² x (or) 

(tan x)' = sec² x

The formula applies to all x where cos(x) ≠ 0

We can derive the derivative of tan x using proofs. To show this, we will use the trigonometric identities along with the rules of differentiation. There are several methods we use to prove this, such as:

• By First Principle
   
• Using Chain Rule
   
• Using Product Rule

We will now demonstrate that the differentiation of tan x results in sec²x using the above-mentioned methods:

The derivative of tan x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. 

To find the derivative of tan x using the first principle, we will consider f(x) = tan x. Its derivative can be expressed as the following limit.

f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = tan x, we write f(x + h) = tan (x + h).

Substituting these into equation (1),

f'(x) = limₕ→₀ [tan(x + h) - tan x] / h

= limₕ→₀ [ [sin (x + h) / cos (x + h)] - [sin x / cos x] ] / h

= limₕ→₀ [ [sin (x + h ) cos x - cos (x + h) sin x] / [cos x · cos(x + h)] ]/ h

We now use the formula sin A cos B - cos A sin B = sin (A - B).

f'(x) = limₕ→₀ [ sin (x + h - x) ] / [ h cos x · cos(x + h)]

= limₕ→₀ [ sin h ] / [ h cos x · cos(x + h)]

= limₕ→₀ (sin h)/ h · limₕ→₀ 1 / [cos x · cos(x + h)]

Using limit formulas, limₕ→₀ (sin h)/ h = 1.

f'(x) = 1 [ 1 / (cos x · cos(x + 0))] = 1/cos² x

As the reciprocal of cosine is secant, we have,

f'(x) = sec² x.

Hence, proved.

To prove the differentiation of tan x using the chain rule,
We use the formula:

Tan x = sin x/ cos x

Consider f(x) = sin x and g (x)= cos x

So we get, tan x = f (x)/ g(x)

By quotient rule: d/dx [f(x) / g(x)] = [f '(x) g(x) - f(x) g'(x)] /  [g(x)]²… (1)

Let’s substitute f(x) = sin x and g (x) = cos x in equation (1),

d/ dx (tan x) = [(cos x) (cos x)- (sin x) (- sin x)]/ (cos x)²
  
 (cos² x + sin² x)/ cos² x …(2)

Here, we use the formula:

(cos² x) + (sin² x) = 1 (Pythagorean identity)

Substituting this into (2),

d/dx (tan x) = 1/ (cos x)²  

Since sec x = 1/cos x, we write:

d/dx(tan x) = sec² x

We will now prove the derivative of tan x using the product rule. The step-by-step process is demonstrated below:

Here, we use the formula,

Tan x = sin x/ cos x

 tan x = (sin x). (cos x)^-1  

Given that, u = sin x and v = (cos x)^-1

Using the product rule formula: d/dx [u.v] = u'. v + u. v'

u' = d/dx (sin x) = cos x.  (substitute u = sin x)

Here we use the chain rule:

v =  (cos x)^-1 = (cos x)^-1 (substitute v = (cos x)^-1)

⇒ v' = -1. (cos)^-2. d/dx (cos x) 

v' = sin x/ (cos x)²  

Again, use the product rule formula:

d/dx (tan x) = u'. v + u. V'

Let’s substitute u = sin x, u' = cos x, v = (cos x)^-1, and v' = sin x/ (cos x)²
 
When we simplify each term:

We get, d/dx (tan x) = 1 + sin²x / (cos x)²

Sin² x/ (cos x)² = tan² x (we use the identity sin² x + cos² x =1)

Thus: 

d/dx (tan x) = 1 + tan² x

Since, 1 + tan² x = sec² x

⇒ d/dx (tan x) = sec² x.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like tan (x).

• For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point.

• The second derivative is derived from the first derivative, which is denoted using f′′ (x)

• Similarly, the third derivative, f′′′(x) is the result of the second derivative and this pattern continues.

• For the nth Derivative of tan(x), we generally use f n(x) for the nth derivative of a function f(x) which tells us the change in the rate of change. (continuing for higher-order derivatives).

Special Cases:

When the x is π/2, the derivative is undefined because tan (x) has a vertical asymptote there.
When the x is 0, the derivative of tan x = sec² (0), which is 1.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Tangent Function: The tangent function is one of the primary six trigonometric functions and is written as tan x.

• Secant Function: A trigonometric function that is the reciprocal of the cosine function. It is typically represented as sec x.

• First Derivative: It is the initial result of a function, which gives us the rate of change of a specific function.

• Asymptote: The function goes near a line without intersecting or crossing it. This line is known as an asymptote.