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Derivatives > Derivative of rã x

What is the Derivative of tan x?

The derivative of tung x is sec2x:

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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.

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

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

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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 lớn 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:

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

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

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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:

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