The derivative of sin x is:

**The reasoning** The derivative tells us the **rate of change** of a function. For sin x, we're asking: "How fast is the sine value changing at any point x?" From first principles (or the standard derivative rules you must memorize): d/dx (sin x) = cos x This is a **fundamental derivative** — one of the building blocks of calculus. Just like how the derivative of x² is 2x, the derivative of sin x is cos x. No negative sign, no other trig function. Pure and simple. **Why the wrong options tempt you** - **−cos x** is the derivative of cos x, not sin x. Students mix these up! - **−sin x** is the derivative of cos x. The pairing gets confusing if you don't practice. - **tan x** looks trigonometric but has nothing to do with differentiating sine. The negative signs in options B and C are traps for students who half-remember the rules. **Quick takeaway** **d/dx (sin x) = cos x** and **d/dx (cos x) = −sin x** — memorize this pair like your name; they're the heartbeat of calculus.