How to Find Derivatives of Trig Functions

Trigonometric derivatives show up constantly in calculus — from basic differentiation quizzes to physics and engineering applications. If you can nail these six core derivatives and learn how to combine them with the chain rule, product rule, and quotient rule, you will be able to handle virtually any trig derivative thrown at you.

This guide walks through every formula you need, five worked examples, and the most common mistakes students make. If you are working through a broader calculus course, check out our Calculus Guide for the full picture, or brush up on identities in our Trigonometry Guide.

The Six Core Trig Derivatives

Memorize these. Everything else builds on them.

Function	Derivative
sin(x)	cos(x)
cos(x)	-sin(x)
tan(x)	sec²(x)
cot(x)	-csc²(x)
sec(x)	sec(x)tan(x)
csc(x)	-csc(x)cot(x)

A helpful pattern: the three “co-” functions (cos, cot, csc) all pick up a negative sign in their derivatives. That single observation cuts your memorization work in half.

Chain Rule with Trig Functions

When the argument of a trig function is anything other than plain x, you need the chain rule. The template is:

d/dx [trig(u)] = (derivative of outer trig) times (du/dx)

Example 1: Derivative of sin(3x²)

d/dx [sin(3x²)] = cos(3x²) d/dx[3x²] = cos(3x²) 6x = 6x cos(3x²)

Example 2: Derivative of tan(5x)

d/dx [tan(5x)] = sec²(5x) * 5 = 5 sec²(5x)

The chain rule is where most errors happen. Always ask yourself: “Is the inside just x?” If not, multiply by the derivative of the inside.

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Product Rule with Trig

When two functions are multiplied together, use the product rule: d/dx [f * g] = f’g + fg’.

Example 3: Derivative of x² sin(x)

Let f = x² and g = sin(x).

d/dx [x² sin(x)] = 2x sin(x) + x² cos(x)

That is the final answer — no further simplification needed unless the problem asks for it.

Quotient Rule with Trig

For a fraction of two functions, use d/dx [f/g] = (f’g - fg’) / g².

Example 4: Derivative of sin(x) / x

Let f = sin(x), g = x.

d/dx [sin(x)/x] = (cos(x) x - sin(x) 1) / x² = (x cos(x) - sin(x)) / x²

Example 5: Derivative of sec(x) / (1 + tan(x))

Let f = sec(x), g = 1 + tan(x).

• f’ = sec(x)tan(x)
• g’ = sec²(x)

d/dx = [sec(x)tan(x)(1 + tan(x)) - sec(x)sec²(x)] / (1 + tan(x))²

Factor out sec(x) from the numerator:

= sec(x)[tan(x)(1 + tan(x)) - sec²(x)] / (1 + tan(x))²

Expand tan(x) + tan²(x) - sec²(x). Since sec²(x) = 1 + tan²(x), this simplifies to tan(x) - 1:

= sec(x)(tan(x) - 1) / (1 + tan(x))²

This kind of simplification is exactly where a free math checker saves you time — plug in your answer and confirm it matches.

Common Mistakes

1. Forgetting the negative sign on co-functions. The derivative of cos(x) is -sin(x), not sin(x). Double-check every time.

2. Dropping the chain rule. d/dx [sin(2x)] is 2cos(2x), not cos(2x). The inner derivative must be included.

3. Mixing up sec and csc derivatives. sec(x) differentiates to sec(x)tan(x). csc(x) differentiates to -csc(x)cot(x). Do not swap the tan and cot.

4. Applying the product rule when you need the chain rule (or vice versa). sin(x) * cos(x) needs the product rule. sin(cos(x)) needs the chain rule. Read the structure of the expression before choosing a rule.

5. Forgetting to simplify with identities. After differentiating, check whether a Pythagorean identity (sin²x + cos²x = 1, 1 + tan²x = sec²x) can clean up the result.

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Quick-Reference Strategy

When you see a trig derivative problem, follow this checklist:

1. Identify the outermost operation — is it a product, quotient, or composition?
2. Pick the right rule (product, quotient, or chain).
3. Recall the base trig derivative from the table above.
4. Differentiate, then simplify using trig identities if possible.

FAQ

Do I need to memorize all six trig derivatives?

Yes, for any timed exam you should have them committed to memory. The negative-sign pattern on the co-functions makes it manageable. In practice, sin, cos, and tan come up far more often than cot, sec, and csc.

When do I use the chain rule versus the product rule?

Use the chain rule when one function is inside another, like sin(x²). Use the product rule when two functions are multiplied, like x sin(x). If you see both situations at once — for example x sin(x²) — apply the product rule first, then the chain rule on the inner composition.

How do I check my derivative answer?

Plug in a simple value like x = 0 or x = pi/4 into both your derivative and a numerical approximation (using the limit definition with a small h). If they match, your answer is likely correct. Or use Math.Photos to verify your work instantly.

What are the derivatives of inverse trig functions?

Those are a separate set of formulas (e.g., d/dx [arcsin(x)] = 1/sqrt(1-x²)). They build on the standard trig derivatives but deserve their own treatment. See our Trigonometry Guide for coverage of inverse trig topics.