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Derivatives > Derivative of rã x

## What is the Derivative of tan x?

The derivative of tung x is sec2x:

**See also:**

## How lớn Take the Derivative of chảy x

You can take the derivative of rã 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) và the denominator (bottom term) h(x).** You could use any names you like, as it won’t make a difference lớn the algebra. However, g(x) & h(x) are common choices.

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

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

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

Which we can rewrite as:f′(x) = cos2(x) + sin2(x) / cos(x)2.Step 4: **Use algebra / trig identities lớn simplify. **

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## Proof of the Derivative of tan x

There are a couple of ways to lớn 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, & 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 tung x is sec2x.

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