The Ultimate 2025 Guide to the Derivatives of a Trig Function
If you’re studying calculus, you’ll quickly run into the derivatives of a trig function. These six (plus inverse) rules appear everywhere — from physics to engineering. In this pillar article, we give you a high-level overview of every key derivative, with links to more detailed guides for each one.
🔑 Key Takeaways
- There are six primary trig derivatives, plus one for arctan.
- Every derivative comes from the limit definition – you don’t have to re-derive them each time.
- Memorization is the fastest path: use flashcards or mnemonics.
- Check your work by graphing the derivative function.
- This hub links to detailed spokes for each formula.
Table of Contents
- Derivatives of Sine and Cosine
- Derivatives involving tan x
- Derivatives of cot, sec, csc
- Derivative of arctan x
- Memory Tricks & Common Mistakes
- FAQ
Derivatives of a Trig Function: Sine and Cosine
The derivatives of a trig function start here. Sine and cosine are the foundation.
What is the derivative of sin?
For the sine function with no argument shown, we assume $x$: $\frac{d}{dx}\sin x = \cos x$. This is the most basic of the derivatives of a trig function. For a full derivation and examples, read the spoke article on what is the derivative of sin.
What is the derivative of sinx?
The notation $\sin x$ is the same as $\sin(x)$. So the derivative is $\cos x$. Master this formula because it reappears in many derivatives of a trig function problems. Visit the spoke article on what is the derivative of sinx for step-by-step examples.
What is the derivative of cos?
$\frac{d}{dx}\cos x = -\sin x$. The negative sign often trips students up. This is a core derivative of a trig function. For deeper practice, check the spoke article what is the derivative of cos.
Derivatives of a Trig Function: Tangent and its Variations
The tangent function and its different notations are a common source of confusion. Here we cover three distinct (but mathematically identical) forms.
What is the derivative of tanx?
$\frac{d}{dx}\tan x = \sec^2 x$. This is one of the most useful derivatives of a trig function. Derive it using the quotient rule. For a full walkthrough, see the spoke article on what is the derivative of tanx.
What is the derivative of tangent?
The function “tangent” is the same as $\tan x$, so its derivative is also $\sec^2 x$. Many calculus textbooks use the word “tangent” instead of “tan” in prose. This derivative of a trig function is essential for related rates problems. Dive deeper in the spoke article on what is the derivative of tangent.
What is the derivative of tan?
When you see just “tan” without $x$, it’s shorthand for $\tan x$. So $\frac{d}{dx}\tan = \sec^2 x$. This is another way to write the same derivative of a trig function. For multiple examples and common pitfalls, read what is the derivative of tan.
“The derivative of tangent is secant squared — memorize it and you’ve solved half the derivative problems in calculus.”
Derivatives of a Trig Function: Cot, Sec, Csc
The remaining three primary trig functions also have their own derivative rules. Let’s break them down.
What is the derivative of cot?
$\frac{d}{dx}\cot x = -\csc^2 x$. Note the negative sign. This derivative of a trig function is the version of the tangent derivative with co-functions. For more detail, see the spoke article what is the derivative of cot.
What is the derivative of sec?
$\frac{d}{dx}\sec x = \sec x\tan x$. The product of secant and tangent appears often. This is a key derivative of a trig function. Visit the spoke article what is the derivative of sec for worked problems.
What is the derivative of secx?
Same as above: $\frac{d}{dx}\sec x = \sec x\tan x$. The notation $\sec x$ is standard. This derivative of a trig function is also derived from the reciprocal identity. Check the spoke article on what is the derivative of secx for additional practice.
What is the derivative of csc?
$\frac{d}{dx}\csc x = -\csc x\cot x$. Like sec, but with a negative sign. This derivative of a trig function is less common but still important. For a full explanation, see what is the derivative of csc.
What is the derivative of cscx?
Again, $\csc x$ is the same as $\csc(x)$, so the derivative is $-\csc x\cot x$. This completes the set of six primary derivatives of a trig function. Check the spoke article what is the derivative of cscx for a detailed derivation.
Derivatives of a Trig Function: Arctan
What is the derivative of arctan?
$\frac{d}{dx}\arctan x = \frac{1}{1+x^2}$. This is the only inverse trig derivative in our spoke set. It’s a crucial derivative of a trig function for integration problems. For a full derivation, read the spoke article what is the derivative of arctan.
Memory Tricks & Common Mistakes
Memorizing all derivatives of a trig function takes practice. Here’s a comparison table to help.
| Function | Derivative | Mnemonic |
|---|---|---|
| sin x | cos x | sine → cosine (positive) |
| cos x | –sin x | cosine → –sine |
| tan x | sec² x | tan → sec² (both start with ‘s’) |
| cot x | –csc² x | cot → –csc² (co- functions get negative) |
| sec x | sec x tan x | secant goes with tangent |
| csc x | –csc x cot x | cosecant with cotangent, negative |
| arctan x | 1/(1+x²) | inverse tangent → 1 over 1 plus x² |
A mistake I often see: students confuse the derivative of $\tan x$ with $\cot x$’s derivative. They are not the same. Use the table above as a quick reference whenever you practice derivatives of a trig function.