Derivatives of Trig Functions

Sine and cosine describe anything that repeats: a swinging pendulum, a spinning wheel, the voltage in a wall socket, the height of a tide. To talk about how fast those things are changing, you need the derivatives of the trig functions. The good news is that there are only six, two of them do all the real work, and they fit together in a very tidy pattern.

The Six Trig Derivatives

Here they are as a chart, with each function next to its co-function:

One condition comes with every line of this chart: is in radians. If you’ve skipped the radians and degrees review, it’s worth a quick look, because the reason for this condition sits right inside the proof of the first rule.

The Derivative of sin x

Before any algebra, look at the graph. At the sine curve climbs at a 45° angle, so its slope is 1. At the top of the hump, , the curve is flat: slope 0. At it’s heading down as steeply as it ever does, slope . Now plot those slopes as heights: 1, then 0, then . That’s the cosine curve. So

The picture makes the rule believable. The proof makes it certain, and it uses the limit definition of the derivative together with the sine addition formula:

Group the terms and split the fraction in two:

Two special limits finish the job. The first, , goes to 0. The second, , goes to 1, which is the famous result proved with the squeeze theorem. That leaves .

Here’s why radians matter. The limit is only true when is measured in radians. In degrees the limit is , and the derivative of would be instead. Radians are the units that make the rule clean.

The Derivative of cos x

The same argument, using the cosine addition formula, gives

Don’t lose the minus sign. Look back at the figure: cosine starts at its peak at and immediately heads down, so its slope just to the right of 0 is negative, while is positive there. A negative slope needs a negative sign.

There’s also a shortcut proof. Cosine is just sine shifted, , so the chain rule gives .

The Sine and Cosine Derivative Cycle

Keep differentiating and something nice happens:

Every fourth derivative brings you back to where you started. That turns a scary-looking question like “find the 27th derivative of ” into a remainder problem: , so go six full laps (which change nothing) and then three steps, landing on . This loop is also why sine and cosine show up in the physics of springs and waves: they’re the functions whose second derivative is their own negative.

The Derivative of tan x

The other four functions are built from sine and cosine, so their derivatives come from the quotient rule. For tangent, write :

The Pythagorean identity does the simplifying. The graph agrees: is never smaller than 1, so the tangent curve rises everywhere, at slope 1 through the origin and more steeply as it climbs toward its asymptotes.

You may also see the answer written as , which is the same thing by the identity .

The Derivative of sec x

Write . The numerator is a constant, so its derivative is 0:

Splitting into is the step that turns an unfamiliar fraction into the standard form.

The Derivatives of csc x and cot x

Cosecant and cotangent work exactly the same way, starting from and :

A Memory Trick for the Trig Derivative Rules

You only need to memorise three derivatives: , and . To get the other three, replace every function with its co-function and put a minus sign in front:

• becomes
• becomes
• becomes

A quick check on an exam: the three functions whose names start with “c” (cosine, cotangent, cosecant) are exactly the three with negative derivatives.

Trig Derivatives With the Chain Rule

In real problems the angle is rarely plain . With an inside function , each rule picks up a factor of from the chain rule:

For example:

The same goes for the reciprocal functions: . Keep the inside unchanged in every factor, and multiply by its derivative once.

Example 1: A Constant Times sin x

Differentiate .

Constants ride along, and each term is differentiated separately:

The second term is where the sign slips happen: minus a negative is a plus.

Example 2: A Product With tan x

Differentiate .

The product rule with and :

Example 3: A Quotient That Simplifies

Differentiate .

Replace with 1. The numerator becomes , which cancels one factor of the denominator:

With trig quotients, always look for in the numerator. It’s usually there on purpose.

Example 4: sin² x

Differentiate .

Read it as : the outer layer is the square, the inner layer is . By the chain rule,

The last step uses the double angle identity . Both forms are correct.

Example 5: The Tangent Line to tan x

Find the equation of the tangent line to at .

The point is , since . The slope is . In point-slope form:

That’s the green line in the tan x graph above.

Example 6: A Higher Derivative

Find the 50th derivative of .

The cycle for cosine is . Since , twelve full laps change nothing and two more steps land on .

Common Mistakes With Trig Derivatives

• Dropping a minus sign. The derivatives of , and are all negative.
• Forgetting the chain rule. is , not .
• Writing for the derivative of . That’s the derivative of . The derivative of is .
• Mixing up and . The first is with derivative ; the second is with derivative .
• Using degrees. On a calculator, check you’re in radian mode before you evaluate a derivative at a number.

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. Differentiate .

Problem 2. Differentiate .

Problem 3. Differentiate .

Problem 4. Differentiate .

Problem 5. Find the slope of at .

With the trig functions done, there are two more families to add to your toolkit: exponentials and logarithms, whose derivatives come next, and the inverse trig functions, which turn out to have derivatives with no trig in them at all.