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 right-angled 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.

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 sec²x. Tan x is differentiable in its domain. To prove the differentiation of tan x to be sec²x, 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) = sec²x (or)
• (tan x)' = sec²x

Now we will prove this in different methods in the upcoming sections.

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/cos²x

We know that the reciprocal of cos is sec. So,

f'(x) = sec²x.

Hence proved.

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) = -csc²x. Also, by a property of exponents, a^-m = 1/a^m.

y' = -1/cot²x · (-csc²x)

y' = tan²x · csc²x

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

y' = (sin²x)/(cos²x) · (1/sin²x)

= 1/cos²x

We have 1/cos x = sec x. So

y' = sec²x

Hence proved.

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)] / (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. 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^-1x are NOT same.
  d/dx(tan x) = sec²x
  d/dx(tan^-1x) = 1/(1 + x²)

Topics Related to Differentiation of Tan x:

Here are some topics that you may be interested in while learning the derivative of tan x.

• Calculus Formulas
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• Derivative Formulas
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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) = sec²x. It can also be written as (tan x)' = sec²x.

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)] / (cos²x) = [cos²x + sin²x] / (cos²x) = 1/(cos²x) = sec²x.

What is the Derivative of Tan x²?

We know that d/dx(tan x) = sec²x. So d/dx(tan x²) = sec²x² d/dx(x²) = 2xsec²x² (by chain rule).

What is the Differentiation of Tan x in Terms of Cos x?

We know that the derivative of tan x is sec²x. Also, sec x = 1/(cos x). So d/d(tan x) = 1/cos²x.

What is the Derivative of tan x^-1?

By using the derivative of tan x and chain rule, d/dx(tan x^-1) = sec²x^-1 d/dx(x^-1) = sec²x^-1 (-1 x^-2) = (-sec²x^-1)/(x²).

Is the Derivative of Tan x Equal to Derivativative of Tan^-1x?

No, the derivatives of tan x and tan^-1x are different. The derivative of tan x is sec²x whereas the derivative of tan⁻¹x is 1/(1 + x²).

What is the Difference Between the Derivative of Tan x and the Antiderivative of Tan x?

The derivative of tan x is sec²x. The antiderivative of tan x is nothing but the integral of tan x and ∫ tan x dx = ln |sec x| + C.