Finding the Derivative of a Trig Function: How to Guide
The derivative of sin x is cos x, the derivative of cos x is −sin x, and the derivative of tan x is sec²x; the remaining three follow from the reciprocal and quotient rules: d/dx csc x = −csc x cot x, d/dx sec x = sec x tan x, and d/dx cot x = −csc²x. Finding the derivative of a trig function comes down to memorizing those six results and then applying the chain, product, or quotient rule when the argument is anything other than a plain x. This guide lists every formula, works through one extended example, and flags the mistakes that cost the most points.
Derivatives of Trig Functions
Every derivative in this family is built from sine and cosine, the two functions whose derivatives cycle on their own: sin x → cos x → −sin x → −cos x → sin x. Because the other four trig functions are defined as ratios of sine and cosine, their derivatives can be derived rather than memorized from scratch.
The six core results:
- d/dx (sin x) = cos x
- d/dx (cos x) = −sin x
- d/dx (tan x) = sec²x
- d/dx (csc x) = −csc x cot x
- d/dx (sec x) = sec x tan x
- d/dx (cot x) = −csc²x
Two patterns make the list easier to hold onto. First, the three “co-” functions — cosine, cosecant, cotangent — all carry a negative sign; their non-co counterparts are positive. Second, the derivatives of the reciprocal pair (sec and csc) each look like the function multiplied by the other function: sec x tan x, csc x cot x. Standard treatments of these identities and their derivations appear in Khan Academy and MIT OpenCourseWare.
One caution before moving on: the formulas above assume the argument is exactly x. If it is anything else, they are only the outer layer of the problem.
Basic Rules and Formulas
When a trig function contains an expression instead of a bare x, or when two functions are multiplied or divided, the differentiation rules come into play. The table below pairs each rule with a trig example.
| Rule | General form | Trig example |
|---|---|---|
| Chain rule | d/dx f(g(x)) = f′(g(x)) · g′(x) | d/dx sin(5x) = cos(5x) · 5 |
| Product rule | d/dx [f·g] = f′g + fg′ | d/dx [x² sin x] = 2x sin x + x² cos x |
| Quotient rule | d/dx [f/g] = (f′g − fg′)/g² | d/dx [sin x / x] = (x cos x − sin x)/x² |
The chain rule is the one you will use most, because trig functions almost never appear alone. Read it as “derivative of the outside, times derivative of the inside.” For sin(5x), the outside is sine, whose derivative is cosine; the inside is 5x, whose derivative is 5. Multiply: 5 cos(5x). Dropping the inner derivative is the single most common error in this topic.
Ask what structure sits in front of you. A function inside a function means chain rule. Two things multiplied means product rule. One thing divided by another means quotient rule. Many problems need two rules stacked.
Worked Example: y = cos(3x²)
This one problem exercises the chain rule with a nonlinear inner function, which is where most sign and coefficient errors happen.
Structure: a trig function whose argument is itself a function, so the chain rule applies — and here the inner derivative does real work.
- Outside derivative: the derivative of cosine is negative sine, so the outer piece is −sin(3x²).
- Inside derivative: d/dx (3x²) = 6x.
- Multiply: y′ = −sin(3x²) · 6x = −6x sin(3x²).
Notice that the negative sign comes from the cosine rule and stays put; the 6x comes from the inner function. Keeping those two sources separate prevents most mistakes. For comparison, a linear inner function behaves the same way: d/dx tan(4x) = 4 sec²(4x), where the 4 is just the inside derivative.
Common Mistakes to Avoid
- Sign confusion. Forgetting the minus on cos x, csc x, or cot x derivatives. The three “co-” functions are negative; the other three are positive.
- Dropping the inner derivative. Writing d/dx sin(3x) = cos(3x) instead of 3 cos(3x). Always multiply by the derivative of the argument.
- Mixing up sec and csc. The derivative of sec x is sec x tan x, and the derivative of csc x is −csc x cot x. Pair each function with its own partner, not the other one.
- Using the power rule on trig functions. d/dx sin²x is not 2 sin x; it requires the chain rule and equals 2 sin x cos x, or sin(2x).
- Applying the product rule to a composition. x sin x is a product, but sin(x²) is a composition — different rules, different answers.
Key Takeaways
- The six core derivatives: sin x → cos x, cos x → −sin x, tan x → sec²x, csc x → −csc x cot x, sec x → sec x tan x, cot x → −csc²x.
- The “co-” functions (cos, csc, cot) produce negative derivatives; the others are positive.
- Use the chain rule whenever the argument is not plain x, and always multiply by the inner derivative.
- Use the product rule for multiplied functions and the quotient rule for divided ones; many problems combine two rules.
- Check every answer for three things: correct sign, included inner derivative, and the right sec/csc pairing.
