Question

The derivative of $\sin x$ is:

• A. $\cos x$
• B. $-\cos x$
• C. $-\sin x$
• D. $\tan x$

Hint:

A helpful mnemonic: Derivatives of "Co-" functions (Cosine, Cotangent, Cosecant) always start with a \textbf{negative} sign. Since Sine is not a "Co-" function, its derivative is positive.

Answer 1

The Correct Option is A

To find the derivative of \( \sin x \), we use the differentiation rules of trigonometric functions. The derivative of the sine function is a fundamental result in calculus. Let's go through the steps:

1. The function given is \( \sin x \).
2. According to the differentiation rule, the derivative of \( \sin x \) with respect to \( x \) is \( \cos x \). This can be written as: \(\frac{d}{dx}(\sin x) = \cos x\)
3. The cosine function is the rate of change of the sine function, which means that at any point on the curve of \( \sin x \), the slope of the tangent is \( \cos x \).

Therefore, the correct answer is: \(\cos x\).

Now, let's briefly look at why the other options are incorrect:

• \(-\cos x\): This would be the derivative if we had negative sine, i.e., \(-\sin x\).
• \(-\sin x\): This is the derivative of \(\cos x\), not \(\sin x\).
• \(\tan x\): This is irrelevant here as it is the derivative of a different function, i.e., \(\tan x\).

In conclusion, the derivative of \( \sin x \) is indeed \( \cos x \).