Derivatives of trigonometric functions
The six trigonometric functions and the three common inverse functions each have one derivative worth memorizing. Everything else on this page — sin(2x), sin²(x), x sin(x) — is one of those nine formulas plus the chain rule or the product rule.
A cofunction picks up a minus sign: the derivative of cos(x) is −sin(x), of cot(x) is −csc²(x), and of csc(x) is −csc(x) cot(x). The inverse functions are a separate family. sin⁻¹(x) means arcsin(x), not 1/sin(x). Their derivatives come from implicit differentiation, not from the quotient rule on a reciprocal.
Derivatives of sin, cos, tan, sec, csc, and cot
Sine and cosine are the base pair. Tangent is sin/cos, so the quotient rule produces sec²(x). Secant, cosecant, and cotangent are the reciprocals, and each derivative keeps the original function in the answer.
| Function | Derivative | Worked page |
|---|---|---|
| sin(x) | cos(x) | Worked pageSolve |
| cos(x) | −sin(x) | Worked pageSolve |
| tan(x) | sec²(x) | Worked pageSolve |
| sec(x) | sec(x) tan(x) | Worked pageSolve |
| csc(x) | −csc(x) cot(x) | Worked pageSolve |
| cot(x) | −csc²(x) | Worked pageSolve |
Derivatives of arcsin, arccos, and arctan
Courses also write these as sin⁻¹(x), cos⁻¹(x), and tan⁻¹(x). The superscript is inverse-function notation. It is not an exponent, so these are not the derivatives of csc, sec, or cot. Arcsin and arccos are only differentiable on (−1, 1). Arctan is differentiable for every real x.
| Function | Derivative | Worked page |
|---|---|---|
| arcsin(x) = sin⁻¹(x) | 1/√(1 − x²) | Worked pageSolve |
| arccos(x) = cos⁻¹(x) | −1/√(1 − x²) | Worked pageSolve |
| arctan(x) = tan⁻¹(x) | 1/(1 + x²) | Worked pageSolve |
Chain rule and product rule on trig functions
sin(2x) is sin of an inside function, so multiply by the derivative of 2x. sin²(x) is a power of sin, so the chain rule contributes 2 sin(x) and the derivative of the inside, cos(x). x sin(x) is a product, so both factors survive.
| Function | Derivative | Worked page |
|---|---|---|
| sin(2x) | 2 cos(2x) | Worked pageSolve |
| cos(2x) | −2 sin(2x) | Worked pageSolve |
| sin²(x) | 2 sin(x) cos(x) | Worked pageSolve |
| cos²(x) | −2 sin(x) cos(x) | Worked pageSolve |
| tan²(x) | 2 tan(x) sec²(x) | Worked pageSolve |
| x sin(x) | sin(x) + x cos(x) | Worked pageSolve |
| x cos(x) | cos(x) − x sin(x) | Worked pageSolve |
Hyperbolic functions students mix up with trig
sinh, cosh, and tanh are not trigonometric. The pattern looks familiar and the signs do not match: the derivative of cosh(x) is sinh(x), with no minus sign. They are listed here because the names are easy to confuse with sin, cos, and tan.
| Function | Derivative | Worked page |
|---|---|---|
| sinh(x) | cosh(x) | Worked pageSolve |
| cosh(x) | sinh(x) | Worked pageSolve |
| tanh(x) | sech²(x) | Worked pageSolve |
Questions about trigonometric derivatives
What is the derivative of sin, cos, and tan?
The derivative of sin(x) is cos(x). The derivative of cos(x) is −sin(x). The derivative of tan(x) is sec²(x). The minus sign on cosine is the one most often dropped.
What is the derivative of sec, csc, and cot?
The derivative of sec(x) is sec(x) tan(x). The derivative of csc(x) is −csc(x) cot(x). The derivative of cot(x) is −csc²(x). Cotangent and cosecant both carry a minus sign.
What is the derivative of the inverse trig functions?
The derivative of arcsin(x), also written sin⁻¹(x), is 1/√(1−x²). The derivative of arccos(x), or cos⁻¹(x), is −1/√(1−x²). The derivative of arctan(x), or tan⁻¹(x), is 1/(1+x²). Each has its own step-by-step page.
Does sin⁻¹(x) mean 1/sin(x)?
No. In calculus, sin⁻¹(x) means the inverse sine, arcsin(x). The reciprocal 1/sin(x) is csc(x), and its derivative is −csc(x) cot(x), not 1/√(1−x²).