How to Differentiate the Trigonometric Functions

You should memorize the derivatives of the six trig functions. Make sure you memorize the first two in the following list — they’re a snap. If you’re good at rote memorization, memorize the last four as well.

Here they are:

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You might enjoy the following mnemonic trick for the last four trig derivatives. (This might seem a bit convoluted, but it works). Imagine you’re taking a test and can’t remember these derivatives. You lean over to the guy next to you and whisper, “Psst, hey, what’s the derivative of cscx?” Now, take the last three letters of psst (sst) — those are the initial letters of sec, sec, tan. Write these three functions down, and below them, write their cofunctions: csc, csc, cot. Put a negative sign on the csc in the middle. Finally, add arrows:

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Using this diagram, the trig derivatives are very easy to remember. Look at the top row. The sec on the left has an arrow pointing to sec tan — so the derivative of secx is secx tanx.

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The bottom row works the same way, except that both derivatives are negative.

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