Calculus > Proof of the Derivative Tan x
The derivative tan x is sec2x:
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 sec2x.
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) + cos2(x) = 1:
Step 5: Substitute the trigonometric identity 1/cos2x=sec2x to get the final answer:
d/dx tan x = sec2x
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