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Problem

Let \[g(x)=\dfrac{\cos(x)}{\sin(x)}\].

**Find \[g'(x)\].**

Choose 1 answer:

Choose 1 answer:

(Choice A)   

\[\dfrac{1}{\sin^2(x)}\]

A

\[\dfrac{1}{\sin^2(x)}\]

(Choice B)   

\[\dfrac{\sin^2(x)-\cos^2(x)}{\sin^2(x)}\]

B

\[\dfrac{\sin^2(x)-\cos^2(x)}{\sin^2(x)}\]

(Choice C)   

\[-\dfrac{1}{\sin^2(x)}\]

C

\[-\dfrac{1}{\sin^2(x)}\]

(Choice D)   

\[\dfrac{\cos^2(x)-\sin^2(x)}{\sin^2(x)}\]

D

\[\dfrac{\cos^2(x)-\sin^2(x)}{\sin^2(x)}\]

To find the derivative \( g'(x) \) of the function \( g(x) = \frac{\cos(x)}{\sin(x)} \), we can use the quotient rule. Recall that the quotient rule is:

\[
\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]

For the function \( g(x) = \frac{\cos(x)}{\sin(x)} \), let:
- \( u(x) = \cos(x) \)
- \( v(x) = \sin(x) \)

Now, compute the derivatives:
- \( u'(x) = -\sin(x) \)
- \( v'(x) = \cos(x) \)

Applying the quotient rule:

\[
g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} = \frac{(-\sin(x))\sin(x) - \cos(x)\cos(x)}{\sin^2(x)}
\]

Simplify the expression in the numerator:

\[
g'(x) = \frac{-\sin^2(x) - \cos^2(x)}{\sin^2(x)}
\]

Using the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \):

\[
g'(x) = \frac{-(\sin^2(x) + \cos^2(x))}{\sin^2(x)} = \frac{-1}{\sin^2(x)}
\]

So the correct derivative is:

\[
g'(x) = -\frac{1}{\sin^2(x)}
\]

This corresponds to **Choice C**.

Would you like any further explanations?

Here are 5 related questions to explore this concept further:

1. How would you find the second derivative \( g''(x) \) of this function?
2. What is the derivative of \( g(x) = \cot(x) \), and how does it relate to this problem?
3. Can you apply the chain rule to simplify \( g'(x) \) in a different form?
4. How does the derivative \( g'(x) \) change at specific values, such as \( x = \frac{\pi}{4} \)?
5. What happens to \( g(x) \) and \( g'(x) \) as \( x \) approaches 0 or \( \pi \)?

**Tip:** Remember that trigonometric identities, like the Pythagorean identity, can simplify your work when dealing with derivatives involving trigonometric functions.

Learn how to find the derivative of g(x) = cos(x) / sin(x) using the quotient rule and trigonometric identities. This problem explores differentiation of trigonometric functions and application of the quotient rule. Suitable for advanced high school students.

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