Calculus > Proof of the Derivative Tan x

The derivative tan x is sec^{2}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, which relies *only *on trig identities and a little algebra, skipping the need for using the definition of a derivative at all.

## Proof of the Derivative Tan x: Steps

**Sample 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!

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