Try the rule calculator
Definition and meaning
The six basic trigonometric derivatives are: (sin x)'=cos x, (cos x)'=-sin x, (tan x)'=sec^2 x, (cot x)'=-csc^2 x, (sec x)'=sec x tan x, and (csc x)'=-csc x cot x. These formulas assume angles measured in radians.
Why the rule works
For sine and cosine, the limit definition combines angle-addition identities with the limits sin(h)/h→1 and [cos(h)-1]/h→0. The tangent formula follows from tan x=sin x/cos x and the quotient rule. Reciprocal identities similarly produce the derivatives of cotangent, secant, and cosecant.
A reliable hand method
Identify the trig function, apply its standard derivative, and then use the chain rule if its angle is anything other than x. Preserve the original angle while differentiating the outer trig function. Simplify with identities only when doing so makes the answer easier to interpret.
Example 1: sin(3*x)
Answer: 3cos(3x)
- Differentiate sine to cosine.
- Multiply by the derivative of 3x.
Example 2: cos(x^2)
Answer: -2x sin(x^2)
- Cosine introduces a minus sign.
- The inner square contributes 2x.
Example 3: tan(5*x-1)
Answer: 5sec^2(5x-1)
- Use the tangent derivative.
- Multiply by the linear inner derivative 5.
Example 4: sec(x)*tan(x)
Answer: sec(x)tan^2(x)+sec^3(x)
- Apply the product rule.
- Use derivatives sec'=sec tan and tan'=sec^2.
Common mistakes
The cosine derivative is negative, while the sine derivative is not. Do not confuse sec^2(x), the derivative of tangent, with sec(x^2). Degree-mode formulas require a conversion factor; standard calculus formulas use radians. Nested angles also require the chain rule.
How to verify an answer
Differentiate the expression independently, compare the structure of each term, and substitute a safe numerical value into both the symbolic derivative and a centered difference quotient. Avoid points outside the original domain and points near a discontinuity. A matching numerical value increases confidence, but it cannot establish equality for every input. Algebraic reasoning and domain analysis remain necessary.
When the calculator and handwritten work differ, first check parentheses, multiplication symbols, the selected variable, and any missing chain-rule factor. Then factor or expand both answers to see whether they are equivalent. The DerivCalc methodology explains why simplification is deliberately conservative.
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Questions about this calculator
Can I use the Trigonometric Derivatives Calculator for homework checks?
Yes. Compare the result and named rules with your own work, then check domain restrictions. The numerical spot-check is a consistency test, not a formal proof.
Why can an equivalent answer look different?
Factoring, expanding, reciprocal notation, and trigonometric identities can produce different-looking expressions with the same values on their shared domain.
Does the calculator replace learning the rule?
No. It is designed to expose the rule and intermediate reasoning. The worked examples explain how to select and apply the method by hand.
Editorial standard: formulas and notation are checked against OpenStax Calculus Volume 1, Chapter 3. Content is written and maintained by SilverCodeLabs. Report a correction with the exact expression and expected result.