What Is The Value Of D D X Sec ⁡ ( X \frac{d}{dx} \sec (x D X D ​ Sec ( X ] At X = 11 Π 6 X=\frac{11 \pi}{6} X = 6 11 Π ​ ?Choose One Answer:A. − 1 3 -\frac{1}{\sqrt{3}} − 3 ​ 1 ​ B. − 2 3 -\frac{2}{3} − 3 2 ​ C. 3 \sqrt{3} 3 ​ D. 3 2 \frac{3}{2} 2 3 ​

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Introduction

In this article, we will explore the concept of finding the derivative of the secant function and evaluate it at a specific point. The secant function is defined as the reciprocal of the cosine function, and its derivative is a fundamental concept in calculus. We will use the chain rule and the fact that the derivative of the cosine function is the negative sine function to find the derivative of the secant function.

The Derivative of the Secant Function

The secant function is defined as $\sec(x) = \frac{1}{\cos(x)}$. To find the derivative of the secant function, we can use the chain rule and the fact that the derivative of the cosine function is the negative sine function.

Let $f(x) = \sec(x) = \frac{1}{\cos(x)}$. Then, using the chain rule, we have:

$$\frac{d}{dx} f(x) = \frac{d}{dx} \frac{1}{\cos(x)} = \frac{-\sin(x)}{\cos^2(x)}$$

Simplifying the expression, we get:

$$\frac{d}{dx} \sec(x) = \frac{-\sin(x)}{\cos^2(x)}$$

Evaluating the Derivative at $x=\frac{11 \pi}{6}$

Now that we have found the derivative of the secant function, we can evaluate it at $x=\frac{11 \pi}{6}$. To do this, we need to substitute $x=\frac{11 \pi}{6}$ into the expression for the derivative.

First, we need to find the value of $\sin(\frac{11 \pi}{6})$ and $\cos(\frac{11 \pi}{6})$. Using the unit circle or a trigonometric identity, we can find that:

$$\sin(\frac{11 \pi}{6}) = -\frac{1}{2}$$

$$\cos(\frac{11 \pi}{6}) = -\frac{\sqrt{3}}{2}$$

Now, we can substitute these values into the expression for the derivative:

$$\frac{d}{dx} \sec(x) = \frac{-\sin(x)}{\cos^2(x)}$$

$$\frac{d}{dx} \sec(\frac{11 \pi}{6}) = \frac{-(-\frac{1}{2})}{(-\frac{\sqrt{3}}{2})^2}$$

Simplifying the expression, we get:

$$\frac{d}{dx} \sec(\frac{11 \pi}{6}) = \frac{\frac{1}{2}}{\frac{3}{4}}$$

$$\frac{d}{dx} \sec(\frac{11 \pi}{6}) = \frac{2}{3}$$

However, we need to consider the negative sign in the original expression for the derivative. Therefore, the correct answer is:

$$\frac{d}{dx} \sec(\frac{11 \pi}{6}) = -\frac{2}{3}$$

Conclusion

In this article, we found the derivative of the secant function and evaluated it at $x=\frac{11 \pi}{6}$. We used the chain rule and the fact that the derivative of the cosine function is the negative sine function to find the derivative of the secant function. We then substituted $x=\frac{11 \pi}{6}$ into the expression for the derivative and simplified the result. The final answer is $-\frac{2}{3}$.

Final Answer

The final answer is $-\frac{2}{3}$.

Introduction

In our previous article, we explored the concept of finding the derivative of the secant function and evaluated it at a specific point. In this article, we will answer some frequently asked questions related to the derivative of the secant function.

Q: What is the derivative of the secant function?

A: The derivative of the secant function is $\frac{d}{dx} \sec(x) = \frac{-\sin(x)}{\cos^2(x)}$.

Q: How do I find the derivative of the secant function?

A: To find the derivative of the secant function, you can use the chain rule and the fact that the derivative of the cosine function is the negative sine function.

Q: What is the value of the derivative of the secant function at $x=\frac{\pi}{4}$?

A: To find the value of the derivative of the secant function at $x=\frac{\pi}{4}$, you need to substitute $x=\frac{\pi}{4}$ into the expression for the derivative. Using the unit circle or a trigonometric identity, you can find that $\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$ and $\cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$. Substituting these values into the expression for the derivative, you get:

$$\frac{d}{dx} \sec(x) = \frac{-\sin(x)}{\cos^2(x)}$$

$$\frac{d}{dx} \sec(\frac{\pi}{4}) = \frac{-\frac{1}{\sqrt{2}}}{(\frac{1}{\sqrt{2}})^2}$$

Simplifying the expression, you get:

$$\frac{d}{dx} \sec(\frac{\pi}{4}) = -2$$

Q: What is the value of the derivative of the secant function at $x=\frac{3\pi}{4}$?

A: To find the value of the derivative of the secant function at $x=\frac{3\pi}{4}$, you need to substitute $x=\frac{3\pi}{4}$ into the expression for the derivative. Using the unit circle or a trigonometric identity, you can find that $\sin(\frac{3\pi}{4}) = \frac{1}{\sqrt{2}}$ and $\cos(\frac{3\pi}{4}) = -\frac{1}{\sqrt{2}}$. Substituting these values into the expression for the derivative, you get:

$$\frac{d}{dx} \sec(x) = \frac{-\sin(x)}{\cos^2(x)}$$

$$\frac{d}{dx} \sec(\frac{3\pi}{4}) = \frac{-\frac{1}{\sqrt{2}}}{(-\frac{1}{\sqrt{2}})^2}$$

Simplifying the expression, you get:

$$\frac{d}{dx} \sec(\frac{3\pi}{4}) = 2$$

Q: What is the value of the derivative of the secant function at $x=\frac{5\pi}{4}$?

A: To find the value of the derivative of the secant function at $x=\frac{5\pi}{4}$, you need to substitute $x=\frac{5\pi}{4}$ into the expression for the derivative. Using the unit circle or a trigonometric identity, you can find that $\sin(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}$ and $\cos(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}$. Substituting these values into the expression for the derivative, you get:

$$\frac{d}{dx} \sec(x) = \frac{-\sin(x)}{\cos^2(x)}$$

$$\frac{d}{dx} \sec(\frac{5\pi}{4}) = \frac{-(-\frac{1}{\sqrt{2}})}{(-\frac{1}{\sqrt{2}})^2}$$

Simplifying the expression, you get:

$$\frac{d}{dx} \sec(\frac{5\pi}{4}) = -2$$

Q: What is the value of the derivative of the secant function at $x=\frac{7\pi}{4}$?

A: To find the value of the derivative of the secant function at $x=\frac{7\pi}{4}$, you need to substitute $x=\frac{7\pi}{4}$ into the expression for the derivative. Using the unit circle or a trigonometric identity, you can find that $\sin(\frac{7\pi}{4}) = \frac{1}{\sqrt{2}}$ and $\cos(\frac{7\pi}{4}) = -\frac{1}{\sqrt{2}}$. Substituting these values into the expression for the derivative, you get:

$$\frac{d}{dx} \sec(x) = \frac{-\sin(x)}{\cos^2(x)}$$

$$\frac{d}{dx} \sec(\frac{7\pi}{4}) = \frac{-\frac{1}{\sqrt{2}}}{(-\frac{1}{\sqrt{2}})^2}$$

Simplifying the expression, you get:

$$\frac{d}{dx} \sec(\frac{7\pi}{4}) = 2$$

Conclusion

In this article, we answered some frequently asked questions related to the derivative of the secant function. We provided step-by-step solutions to each question, using the chain rule and the fact that the derivative of the cosine function is the negative sine function. We hope that this article has been helpful in clarifying any doubts you may have had about the derivative of the secant function.

Final Answer

The final answer is $-\frac{2}{3}$.