Derivative of Tan x

Derivative of Tan x is sec²x. Derivative of Tan x refers to the process of finding the change in the tangent function with respect to the independent variable. Derivative of tan x is also known as differentiation of tan x.

In this article, we will learn about the derivative of Tan x and its formula including the proof of the formula using the first principle of derivatives, quotient rule, and chain rule as well.

What is Derivative of Tan x?

Among the trig derivatives, the derivative of the tan x is one of the derivatives. The derivative of the tan x is sec²x The derivative of tan x is the rate of change with respect to angle i.e. x. The resultant of the derivative of tan x is sec²x

Derivative of tan x Formula

The formula for the derivative of tan x is given by:

(d/dx) [tan x] = sec²x

or

(tan x)’ = sec²x

Proof of Derivative of tan x

The derivative of tan x can be proved using the following ways:

• By using the First Principle of Derivative
• By using Quotient Rule
• By using Chain Rule

Derivative of tan x by First Principle

To prove derivative of tan x using First Principle of Derivative, we will use basic limits and trigonometric formulas which are listed below

• tan x = sin x/cos x
• sin(A+B) = sinAcosB+cosAsinB

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

Since f(x) = tan x, we have f(x + h) = tan (x + h).

Substituting these in (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 know that 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)]

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

f'(x) = 1 [ 1 / (cos x · cos(x + 0))] = 1/cos2x

since, reciprocal of cos is sec. Therefore

f'(x) = sec²x.

Hence proved.

Derivative of Tan x Proof by Quotient Rule

In this we will apply quotient rule of derivative to find the formula of the derivative of tan x.

We know that

tan x = (sin x)/(cos x).

So we assume that y = (sin x)/(cos x). Then by quotient rule,

y’ = [ cos x · d/dx (sin x) – sin x · d/dx (cos x)] / (cos²x)

= [cos x · cos x – sin x (-sin x)] / (cos²x)

= [cos²x + sin²x] / (cos²x)

By one of the Pythagorean identities, cos²x + sin²x = 1. So

y’ = 1 / (cos²x) = sec²x

Hence proved.

Derivative of Tan x Proof by Chain Rule

In this method we will find the derivative of tan x using chain rule of derivative

For this let us assume y = tan x as y = 1 / (cot x) = (cot x)^-1. Now, by using power rule and chain rule,

y’ = (-1) (cot x)^-2 · d/dx (cot x)

We have d/dx (cot x) = -cosec²x. Also, by a property of exponents, a^-m = 1/a^m.

y’ = -1/cot²x · (-cosec²x)

y’ = tan2x · cosec²x

Now, tan x = (sin x)/(cos x) and cosec x = 1/(sin x). So

y’ = (sin²x)/(cos²x) · (1/sin²x)

y’ = 1/cos²x

We have 1/cos x = sec x. So

y’ = sec²x

Hence proved.

Solved Examples on Derivative of Tan x

Some examples related to Derivative of Tan x are,

Example 1: Find the derivative of tan²x

Solution:

Let f(x) = tan²x = (tan x)²

By using power rule and chain rule,

f'(x) = 2 tan x.d/dx(tan x)

We know that the derivative of tan x is sec²x

f'(x) = 2 tan x · sec²x

Hence, derivative of the given function is 2 tan x·sec²x

Example 2: Differentiate tan x with respect to sec x.

Solution:

Let us assume v = tan x and u = sec x. Then dv/dx = sec2x and du/dx = sec x · tan x.

We have to find dv/du. We can write this as

dv/du = (dv/dx) / (du/dx)

= (sec²x) / (sec x·tan x)

= (sec x) / (tan x)

= (1/cos x) / (sin x/cos x)

= 1/sin x

= cosec x

Hence, derivative of tan x with respect to sec x is cos x.

Example 3: Find the derivative of tan x·sec²x

Solution:

Let f(x) = tan x·sec2x.

By product rule,

f'(x) = tan x·d/dx (sec²x) + sec²x · d/dx(tan x)

= tan x.(2 sec x) d/dx (sec x) + sec²x (sec²x) (by chain rule)

= 2 sec x tan x (sec x tan x) + sec⁴x

= 2 sec²x tan²x + sec⁴x

Hence, derivative of the given function is 2sec²x tan²x + sec⁴x

Practice Questions on Derivative of Tan x

Various problems related to Derivative of Tan x are,

Q1. Find the derivative of tan(3x)

Q2. Find the derivative of tan 2x

Q3. Evaluate: {d}/{dx} tan(x² + 1)

Q4. Evaluate the derivative of tan x.sin x

Q5. Find: (tan x)².sin x

Derivative of Tan x – FAQs

What is Derivative in Math?

In mathematics derivative of a function is defined as the rate of change of the function with respect to an independent variable.

What is Derivative of Tan x?

The derivative of tan x is sec²x.

What are Different Methods to Prove the Derivative of Tan x

The different methods to prove the derivative of sin x are:

• By using the First Principle of Derivative
• By Quotient Rule
• By Chain Rule

Is Derivative of Tan x and differentiation of tan x same thing?

Yes, the terms derivative and differentiation are synonymous to each other hence, derivative of tan x and differentiation of tan x are the same and equal to sec²x

What is Antiderivative of tan x?

Antiderivative of tan x is integral of tan x which is equal to log |sec x| + c

What is the Second Order Derivative of tan x?

Second Order Derivative of tan x is d²/dx²(tan x) = 2sec²x.tanx