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Derivative of Tan x
The derivative of tan x is the square of sec x. Before proving this, let us recollect some facts about tan x. Tan x in a rightangled triangle is the ratio of the opposite side of x to the adjacent side of x and thus it can be written as (sin x)/(cos x). We use this in doing the differentiation of tan x.
Let us learn the differentiation of tan x along with its proof in different methods and also we will solve a few examples using the derivative of tan x.
What is the Derivative of Tan x?
The derivative of tan x with respect to x is denoted by d/dx (tan x) (or) (tan x)' and its value is equal to sec^{2}x. Tan x is differentiable in its domain. To prove the differentiation of tan x to be sec^{2}x, we use the existing trigonometric identities and existing rules of differentiation. We can prove this in the following ways:
 Proof by first principle
 Proof by chain rule
 Proof by quotient rule
Derivative of Tan x Formula
The formula for differentiation of tan x is,
 d/dx (tan x) = sec^{2}x (or)
 (tan x)' = sec^{2}x
Now we will prove this in different methods in the upcoming sections.
Derivative of Tan x Proof by First Principle
To find the derivative of tan x, we assume that f(x) = tan x. Then by first principle, its derivative is given by the following limit.
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
By sum and difference formulas, 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/cos^{2}x
We know that the reciprocal of cos is sec. So,
f'(x) = sec^{2}x.
Hence proved.
Derivative of Tan x Proof by Chain Rule
We will prove the differentiation of tan x formula by chain rule. For this let us note that we can write y = tan x as y = 1 / (cot x) = (cot x)^{1}. Now, by power rule and chain rule,
y' = (1) (cot x)^{2} · d/dx (cot x)
We have d/dx (cot x) = csc^{2}x. Also, by a property of exponents, a^{m} = 1/a^{m}.
y' = 1/cot^{2}x · (csc^{2}x)
y' = tan^{2}x · csc^{2}x
Now, tan x = (sin x)/(cos x) and csc x = 1/(sin x). So
y' = (sin^{2}x)/(cos^{2}x) · (1/sin^{2}x)
= 1/cos^{2}x
We have 1/cos x = sec x. So
y' = sec^{2}x
Hence proved.
Derivative of Tan x Proof by Quotient Rule
We can apply the quotient rule to derive the formula of the derivative of tan x. For this, we have to write tan x as a fraction. 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)] / (cos^{2}x)
= [cos x · cos x  sin x (sin x)] / (cos^{2}x)
= [cos^{2}x + sin^{2}x] / (cos^{2}x)
By one of the Pythagorean identities, cos^{2}x + sin^{2}x = 1. So
y' = 1 / (cos^{2}x) = sec^{2}x
Hence proved. This proof is the easiest one among all the other proofs of the derivatives of tan x.
Common Misconceptions Related to Derivative of Tan x:
Here is some clarity about some common misconceptions regarding the differentiation of tan x.
 d/dx (tan x) is NOT equal to d/dx(sin x) / d/dx (cos x). Instead, we have to use the quotient rule to find the derivative of tan x (by writing it as (sin x)/(cos x)).
 d/dx (tan x) is NOT cot x. cot x is just the reciprocal of tan x.
 The derivatives of tan x and tan^{1}x are NOT same.
d/dx(tan x) = sec^{2}x
d/dx(tan^{1}x) = 1/(1 + x^{2})
Topics Related to Differentiation of Tan x:
Here are some topics that you may be interested in while learning the derivative of tan x.
Solved Examples Using Derivative of Tan x

Example 1: Find the derivative of tan^{2}x.
Solution:
Let f(x) = tan^{2}x = (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 sec^{2}x. So
f'(x) = 2 tan x · sec^{2}x
Answer: The derivative of the given function is 2 tan x · sec^{2}x.

Example 2: What is the derivative of tan x with respect to sec x.
Solution:
Let v = tan x and u = sec x. Then dv/dx = sec^{2}x and du/dx = sec x · tan x.
We have to find dv/du. We can write this as
dv/du = (dv/dx) / (du/dx)
= (sec^{2}x) / (sec x · tan x)
= (sec x) / (tan x)
= (1/cos x) / (sin x/cos x)
= 1/sin x
= csc x
Answer: The derivative of tan x with respect to sec x is csc x.

Example 3: Find the derivative of tan x · sec^{2}x.
Solution:
Let f(x) = tan x · sec^{2}x.
By product rule,
f'(x) = tan x · d/dx (sec^{2}x) + sec^{2}x · d/dx(tan x)
= tan x · (2 sec x) d/dx (sec x) + sec^{2}x (sec^{2}x) (by chain rule)
= 2 sec x tan x (sec x tan x) + sec^{4}x
= 2 sec^{2}x tan^{2}x + sec^{4}x
Answer: The derivative of the given function is 2 sec^{2}x tan^{2}x + sec^{4}x.
FAQs on Derivative of Tan x
What is the Derivative of Tan x with Respect to x?
The derivative of tan x with respect to x is the square of sec x. i.e., d/dx(tan x) = sec^{2}x. It can also be written as (tan x)' = sec^{2}x.
How to Find the Derivative of Tan x Formula?
Let y = tan x. We have tan x = sin x/cos x. By quotient rule, y' = [ cos x · d/dx (sin x)  sin x · d/dx (cos x)] / (cos^{2}x) = [cos^{2}x + sin^{2}x] / (cos^{2}x) = 1/(cos^{2}x) = sec^{2}x.
What is the Derivative of Tan x^{2}?
We know that d/dx(tan x) = sec^{2}x. So d/dx(tan x^{2}) = sec^{2}x^{2} d/dx(x^{2}) = 2xsec^{2}x^{2 }(by chain rule).
What is the Differentiation of Tan x in Terms of Cos x?
We know that the derivative of tan x is sec^{2}x. Also, sec x = 1/(cos x). So d/d(tan x) = 1/cos^{2}x.
What is the Derivative of tan x^{1}?
By using the derivative of tan x and chain rule, d/dx(tan x^{1}) = sec^{2}x^{1} d/dx(x^{1}) = sec^{2}x^{1} (1 x^{2}) = (sec^{2}x^{1})/(x^{2}).
Is the Derivative of Tan x Equal to Derivativative of Tan^{1}x?
No, the derivatives of tan x and tan^{1}x are different. The derivative of tan x is sec^{2}x whereas the derivative of tan⁻¹x is 1/(1 + x^{2}).
What is the Difference Between the Derivative of Tan x and the Antiderivative of Tan x?
The derivative of tan x is sec^{2}x. The antiderivative of tan x is nothing but the integral of tan x and ∫ tan x dx = ln sec x + C.
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