The derivative of \(\sin(x)\) with respect to \(x\) is \(\cos(x)\).

This result is derived from the limit definition of the derivative. Using the definition:

\[
f'(x) = \lim_{{h \to 0}} \frac{\sin(x+h) - \sin(x)}{h}
\]

Applying the sum-to-product identities and taking the limit, we get:

\[
\frac{\sin(x+h) - \sin(x)}{h} = \frac{2 \cos\left(x + \frac{h}{2}\right) \sin\left(\frac{h}{2}\right)}{h}
\]

As \(h\) approaches 0, \(\frac{\sin\left(\frac{h}{2}\right)}{\frac{h}{2}} \to 1\) and \(\cos\left(x + \frac{h}{2}\right) \to \cos(x)\). Thus:

\[
\lim_{{h \to 0}} \frac{2 \cos\left(x + \frac{h}{2}\right) \sin\left(\frac{h}{2}\right)}{h} = \cos(x)
\]

So, the derivative of \(\sin(x)\) is \(\cos(x)\).

Would you like more details or have any questions?

Here are 5 questions you might ask next:
1. What is the derivative of \(\cos(x)\)?
2. How do you find the derivative of a product of two functions?
3. Can you explain the chain rule in differentiation?
4. What is the second derivative of \(\sin(x)\)?
5. How do you find the derivative of \(\tan(x)\)?

**Tip:** When finding the derivative of trigonometric functions, remember the basic derivatives of \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\) to simplify your calculations.

Explore the calculation process for finding the derivative of sin(x). Learn about the limit definition of derivatives and how sum-to-product identities are applied. This explanation is suitable for college-level students and anyone interested in calculus and trigonometry.

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