What is the derivative of sin(x)? A) cos(x) B) -sin(x) C) -cos(x) D) tan(x)

The correct answer is A) cos(x).

Let me explain the derivative of sin(x):

The derivative of sin(x) is cos(x). This is a fundamental derivative rule in calculus that is often memorized by students.

Key points: - When you take the derivative of sin(x), you get cos(x) - This can be proven using the definition of the derivative and limit laws - It's part of the standard derivative rules for trigonometric functions

So, if you were to differentiate sin(x) with respect to x, the result would be cos(x).

Example: d/dx [sin(x)] = cos(x)

The other options are incorrect: B) -sin(x) is not the correct derivative C) -cos(x) is also not correct D) tan(x) is the tangent function, not the derivative of sine

Let me help you with this calculus question.

The derivative of sin(x) is cos(x).

Therefore, the correct answer is A) cos(x)

To help you understand why: 1. This is one of the fundamental derivatives in calculus 2. You can verify this using: - The limit definition of the derivative - The fact that the slope of the sine function at x=0 is 1, which is the value of cos(0) - The graphical relationship between sine and cosine functions

Here's a helpful way to remember this: - The derivative of sin(x) is cos(x) - The derivative of cos(x) is -sin(x) These two functions continue this pattern, cycling through these derivatives with alternating negative signs.

The other options are incorrect because: B) -sin(x) is the derivative of cos(x), not sin(x) C) -cos(x) is the derivative of sin(x) with respect to -x D) tan(x) is not the derivative of any basic trigonometric function

The derivative of sin(x) is cos(x).

So the correct answer is A) cos(x).

To explain why: When we differentiate sin(x) with respect to x using the rules of calculus, we get cos(x). This is one of the fundamental derivative formulas in trigonometry.

The other options are incorrect: - B) -sin(x) is actually the derivative of cos(x) - C) -cos(x) is the derivative of sin(-x) or -sin(x) - D) tan(x) is not the derivative of any of the basic trig functions (its own derivative is sec²(x))

The derivative of sin(x) is A) cos(x).

This is one of the fundamental derivatives in calculus:

d/dx[sin(x)] = cos(x)

To remember this and related derivatives: - d/dx[sin(x)] = cos(x) - d/dx[cos(x)] = -sin(x) - d/dx[tan(x)] = sec²(x)

So the correct answer is A) cos(x).

The derivative of sin(x) is A) cos(x).

This is one of the fundamental derivatives in calculus. When you differentiate sin(x) with respect to x, you get cos(x).

To remember this, you can think of the pattern of trigonometric derivatives: - d/dx[sin(x)] = cos(x) - d/dx[cos(x)] = -sin(x) - d/dx[tan(x)] = sec²(x)

The derivative of sine gives you cosine, and the derivative of cosine gives you negative sine. This creates a cycle that's useful to memorize!

The derivative of sin(x) is cos(x).

Answer: A) cos(x)

This is one of the fundamental derivatives in calculus. Here's why:

Using the limit definition or standard differentiation rules: - d/dx[sin(x)] = cos(x)

For reference, here are some related basic trigonometric derivatives: - d/dx[sin(x)] = cos(x) - d/dx[cos(x)] = -sin(x) - d/dx[tan(x)] = sec²(x)

So the correct answer is A) cos(x).

The correct answer is A) cos(x).

The derivative of sin(x) with respect to x is:

d/dx[sin(x)] = cos(x)

This is one of the fundamental derivatives in calculus that's worth memorizing.

Here are some related derivatives you might find useful: - d/dx[cos(x)] = -sin(x) - d/dx[tan(x)] = sec²(x) - d/dx[-sin(x)] = -cos(x)

So the answer to your question is A) cos(x) ✓

The derivative of (\sin(x)) is (\cos(x)).

Correct answer: A) (\cos(x))

Explanation:

The derivative of (\sin(x)) with respect to (x) is a fundamental result in calculus. Here's why:

1. Definition of the derivative:
   The derivative of a function (f(x)) is given by:
   [ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]
   Applying this to (\sin(x)), we get:
   [ \frac{d}{dx} \sin(x) = \lim_{h \to 0} \frac{\sin(x + h) - \sin(x)}{h} ]

2. Using trigonometric identities:
   The sine addition formula is:
   [ \sin(x + h) = \sin(x)\cos(h) + \cos(x)\sin(h) ]
   Substituting this into the limit gives:
   [ \frac{d}{dx} \sin(x) = \lim_{h \to 0} \frac{\sin(x)\cos(h) + \cos(x)\sin(h) - \sin(x)}{h} ]
   Simplifying:
   [ = \lim_{h \to 0} \left( \sin(x) \frac{\cos(h) - 1}{h} + \cos(x) \frac{\sin(h)}{h} \right) ]

3. Evaluating the limits:
   - (\lim_{h \to 0} \frac{\sin(h)}{h} = 1) (a standard limit)
   - (\lim_{h \to 0} \frac{\cos(h) - 1}{h} = 0) (another standard limit)

Thus, the derivative simplifies to:
[ \frac{d}{dx} \sin(x) = \sin(x) \cdot 0 + \cos(x) \cdot 1 = \cos(x) ]

So, the correct choice is A) (\cos(x)).

Additional Notes:

• The derivative of (\cos(x)) is (-\sin(x)) (not (\sin(x))).
• The derivative of (\tan(x)) is (\sec^2(x)).
• Memorizing these basic derivatives is helpful for solving more complex problems in calculus!

Let me know if you'd like further clarification. 😊

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

Explanation:

• The derivative measures the rate of change of a function. For (\sin(x)), this follows from the standard trigonometric differentiation rule: [ \frac{d}{dx} \sin(x) = \cos(x) ]
• This result can be derived using the limit definition of the derivative or confirmed via calculus identities.

Answer:

A) (\cos(x)) ✅

Why not the others?

• B) (-\sin(x)): This is the derivative of (\cos(x)), not (\sin(x)).
• C) (-\cos(x)): This is the derivative of (\sin(x)) evaluated at specific points (e.g., under a phase shift), but not the general rule.
• D) (\tan(x)): This is unrelated; (\tan(x) = \frac{\sin(x)}{\cos(x)}), and its derivative is (\sec^2(x)).

Let me know if you’d like a deeper dive into the proof! 😊

The derivative of sin(x) is A) cos(x).

Hello there!

The correct answer is A) cos(x). The derivative of sin(x) is indeed cos(x).

So, the answer is:

A) cos(x)

Happy to help! Let me know if you have any more questions. 😊

Hello! The correct answer is A) cos(x).

The derivative of sin(x) is cos(x). This is a fundamental concept in calculus.

Ahoj! Ráda ti pomůžu s touto matematickou otázkou.

Derivace funkce sin(x) je cos(x).

Takže správná odpověď je A) cos(x).

Přeji hezký den!

Hello there! That's a great calculus question!

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

So, the correct option is A) cos(x).

If you'd like to know why, or perhaps delve into the derivatives of other trigonometric functions, just let me know! Happy to help! 😊

The derivative of (\sin(x)) is:

A) (\cos(x))

So, the correct answer is A) cos(x).

The derivative of (\sin(x)) is (\cos(x)).

So the correct answer is: A) cos(x).

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

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

The derivative of sin(x) with respect to x is A) cos(x). So, the correct answer is A) cos(x).

The derivative of ( \sin(x) ) is ( \cos(x) ). Therefore, the correct answer is:

A) cos(x).

If you have any more questions about calculus or anything else, feel free to ask!

A) cos(x)

Because d/dx[sin(x)] = cos(x).

A) cos(x)

Reason: d/dx[sin x] = cos x (can be shown using the limit definitions and the identities lim_{h→0} (sin h)/h = 1 and lim_{h→0} (1−cos h)/h = 0).

Answer: A) cos(x)

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

The derivative of sin(x) is cos(x). So, the correct answer is A) cos(x).

If you're interested in visualizing this function or its derivative, remember there's an "image generation" option in the input form that you can turn on!

The derivative of (\sin(x)) with respect to (x) is (\cos(x)). Therefore, the correct answer is:

A) (\cos(x))

If you're interested in visualizing this function or its derivative, remember there's an option called "image generation" in the input form that you can turn on to generate related images!

I'm happy to help with this! The derivative of sin(x) is cos(x). So, the correct answer is A) cos(x).

To explain briefly, in calculus, the derivative of a function measures how it changes as its input changes. For trigonometric functions like sin(x), the derivative is found using standard rules. The derivative of sin(x) is cos(x), which reflects the rate of change of the sine function at any point x. If you'd like a deeper explanation or a step-by-step derivation, just let me know!

Hello! I'd be happy to help with your calculus question. The derivative of sin(x) is indeed A) cos(x).

Quick Explanation:

In calculus, the derivative of the sine function, sin(x), with respect to x is cos(x). This is a fundamental rule from differential calculus, often derived using the limit definition or trigonometric identities. For example: - If you have a function f(x) = sin(x), then f'(x) = cos(x).

Why the other options are incorrect:

• B) -sin(x): This is actually the derivative of cos(x), not sin(x).
• C) -cos(x): This is the derivative of sin(x) plus π/2 (or an equivalent phase shift), but it's not the basic derivative of sin(x).
• D) tan(x): This is the derivative of -ln|cos(x)| or related functions, but not sin(x).

If you're studying derivatives, it's a great topic—keep practicing, and feel free to ask if you have more questions or need help with related concepts like the chain rule or integrals! 😊

The derivative of sin(x) is A) cos(x).

To explain briefly: In calculus, the derivative of the sine function with respect to x is the cosine function. This is a standard result from differentiation rules. If you have a more specific context or need a step-by-step derivation (e.g., using the limit definition), feel free to provide more details! 😊

The derivative of sin(x) with respect to x is cos(x). Therefore, the correct choice is A) cos(x).

Mathematically: d/dx [sin(x)] = cos(x).

The derivative of sin(x) is cos(x).
Correct choice: A) cos(x)

The derivative of sin(x) is cos(x).

To explain briefly, the derivative of a function measures how the function's value changes as the input changes. For sin(x), using basic rules of differentiation, we have:

d/dx [sin(x)] = cos(x)

Among the choices given, that corresponds to:

A) cos(x).

The correct answer is A) cos(x).