Every trig derivative, in one place.

The six trig functions, their inverses and the hyperbolics, with the pattern that makes them easy to remember, worked examples, and flashcards to lock them in.

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The six trig functions

Learn the first two cold. The other four follow from them with the quotient rule.

sine

cosine

tangent

cotangent

secant

cosecant

The pattern: every function that starts with “co” has a minus sign in its derivative. The ones without it (sin, tan, sec) don’t. And each derivative uses the function’s co-partner or its own family: sec and csc both give back “itself × the tangent-type partner”.

Inverse trig functions

None of these contain a trig function. They are algebraic. Again, the “co” versions are just the negatives.

arcsine

arccosine

arctangent

arccotangent

arcsecant

arccosecant

Hyperbolic functions, for completeness:

sinh

cosh

tanh

With the chain rule

When the inside isn’t just x, multiply by the derivative of the inside, u′.

Worked examples

Product rule:

Why they’re true

You don’t have to memorise all twelve if you know where they come from.

Start with two limits

From the unit circle and the squeeze theorem:

Plug those into the definition of the derivative with the angle-sum formula and you get sin′ = cos and cos′ = −sin.

The rest are a quotient away

Cotangent, secant and cosecant work the same way, and each inverse comes from implicit differentiation of on a right triangle.

Flashcards

Say the answer to yourself, then tap the card.