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.

Also Check

• Derivative of Inverse Trigonometric Functions
• Differentiation Formulas
• Derivative of Trigonometric Functions

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.