Derivative of Tan x & Proof in Easy Steps

Derivatives > Derivative of Tan x

What is the Derivative of Tan x?

The derivative of tan x is sec2x:
derivative of tan x


See also: Common derivative rules.

How to Take the Derivative of Tan x

You can take the derivative of tan x using the quotient rule. That’s because of a basic trig identity, which happens to be a quotient:

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

Step 1: Name the numerator (top term) in the quotient g(x) and the denominator (bottom term) h(x). You could use any names you like, as it won’t make a difference to the algebra. However, g(x) and h(x) are very common choices.

  • g(x) = sin(x)
  • h(x) = cos(x)

Step 2: Put g(x) and h(x) into the quotient rule formula.

quotient rule

Formal definition for the quotient rule.







Note that I used d/dx here to denote a derivative instead of g(x)’ or h(x)’. You can use either notation: it’s a personal choice.

Step 3: Differentiate the functions from Step 2. There’s two parts to differentiate:

  1. The derivative of the first part of the function—sin(x)— is cos(x)
  2. The derivative of cos(x) is -sin(x).

Placing those derivatives into the equation, we get:
f'(x) = [cos(x)*cos(x) – (-sin(x))*sin(x)] / cos(x)2
f'(x)= cos2(x) + sin2(x) / cos(x)2.

Step 4: Use algebra / trig identities to simplify.

  • Specifically, start by using the identity cos2(x) + sin2(x) = 1
  • This gives you 1/cos2(x), which is equivalent in trigonometry to sec2(x).

If algebra isn’t your strong point, you might find these steps a bit tricky. Chegg offers 30 minutes free tutoring, so if you’re stuck, give them a try!

Proof of the Derivative of Tan x

There are a couple of ways to prove the derivative tan x. You could start with the definition of a derivative and prove the rule using trigonometric identities. But there’s actually a much easier way, and is basically the steps you took above to solve for the derivative. As it relies only on trig identities and a little algebra, it is valid as a proof. Plus, it skips the need for using the definition of a derivative at all.

Steps

Sample problem: Prove the derivative tan x is sec2x.

Step 1: Write out the derivative tan x as being equal to the derivative of the trigonometric identity sin x / cos x:
proof of the derivative tan x


Step 2: Use the quotient rule to get:
-tan-x-5


Step 3: Use algebra to simplify:
tan-x-6


Step 4: Substitute the trigonometric identity sin(x) + cos2(x) = 1:
-tan-x-7


Step 5: Substitute the trigonometric identity 1/cos2x=sec2x to get the final answer:
d/dx tan x = sec2x
That’s it!

References

Nicolaides, A. (2007). Pure mathematics: Differential calculus and applications, Volume 4. Pass Publications.