Derivatives > Derivative of Tan x

## What is the Derivative of Tan x?

The derivative of tan x is sec^{2}x:

**See also:**

## 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 is a quotient of the sine function and cosine function:

**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 common choices.

- g(x) = sin(x)
- h(x) = cos(x)

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

Note that I used d/dx here to denote a derivative (Leibniz Notation) instead of g(x)′ or h(x)′ (Prime Notation (Lagrange), Function & Numbers). You can use either notation: they mean the same thing.

Step 3: ** Differentiate the functions from Step 2.** There are two parts to differentiate:

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

Placing those derivatives into the formula from Step 3, we get:

Which we can rewrite as:

f′(x) = cos^{2}(x) + sin^{2}(x) / cos(x)^{2}.

Step 4: **Use algebra / trig identities to simplify. **

- Specifically, start by using the identity cos
^{2}(x) + sin^{2}(x) = 1 - This gives you 1/cos
^{2}(x), which is equivalent in trigonometry to sec^{2}(x).

## 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

**Example problem**: Prove the derivative tan x is sec^{2}x.

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

Step 2: **Use the quotient rule **to get:

Step 3: **Use algebra** to simplify:

Step 4: **Substitute the trigonometric identity **sin(x) + cos ^{2}(x) = 1:

Step 5: **Substitute the trigonometric identity** 1/cos^{2}x = sec^{2}x to get the final answer:

d/dx tan x = sec^{2}x

*That’s it!
*

## References

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