Let $g(x) = X \cos(x)$.Find $g^{\prime}(x)$.Choose One Answer:A. $\cos(x) - X \sin(x)$B. $ Cos ⁡ ( X ) + X Sin ⁡ ( X ) \cos(x) + X \sin(x) Cos ( X ) + X Sin ( X ) [/tex]C. $-\sin(x)$D. $-\cos(x)$

Introduction

In this article, we will explore the concept of finding the derivative of a trigonometric function. Specifically, we will examine the function $g(x) = x \cos(x)$ and determine its derivative, $g^{\prime}(x)$. We will use the product rule of differentiation to find the derivative of this function.

The Product Rule of Differentiation

The product rule of differentiation states that if we have a function of the form $f(x) = u(x)v(x)$, then the derivative of this function is given by:

$$f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$$

This rule can be applied to any function that can be written as the product of two functions.

Applying the Product Rule to $g(x)$

In this case, we have $g(x) = x \cos(x)$. We can see that this function is the product of two functions: $x$ and $\cos(x)$. Therefore, we can apply the product rule to find the derivative of $g(x)$.

Let $u(x) = x$ and $v(x) = \cos(x)$. Then, we have:

$$u^{\prime}(x) = 1$$

$$v^{\prime}(x) = -\sin(x)$$

Now, we can apply the product rule to find the derivative of $g(x)$:

$$g^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$$

$$g^{\prime}(x) = (1)(\cos(x)) + (x)(-\sin(x))$$

$$g^{\prime}(x) = \cos(x) - x \sin(x)$$

Conclusion

In this article, we used the product rule of differentiation to find the derivative of the function $g(x) = x \cos(x)$. We found that the derivative of this function is $g^{\prime}(x) = \cos(x) - x \sin(x)$. This is the correct answer, which is option A.

Answer

The correct answer is:

A. $\cos(x) - x \sin(x)$

Discussion

This problem is a great example of how to apply the product rule of differentiation to find the derivative of a trigonometric function. The product rule is a powerful tool that can be used to find the derivative of any function that can be written as the product of two functions.

In this case, we had to use the product rule to find the derivative of the function $g(x) = x \cos(x)$. We found that the derivative of this function is $g^{\prime}(x) = \cos(x) - x \sin(x)$.

This problem is also a great example of how to use trigonometric identities to simplify the derivative of a function. In this case, we used the identity $\sin(x) = -\sin(x)$ to simplify the derivative of the function.

Final Thoughts

In conclusion, this problem is a great example of how to apply the product rule of differentiation to find the derivative of a trigonometric function. We found that the derivative of the function $g(x) = x \cos(x)$ is $g^{\prime}(x) = \cos(x) - x \sin(x)$. This is the correct answer, which is option A.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Additional Resources

• [1] Khan Academy: Derivatives
• [2] MIT OpenCourseWare: Calculus

Related Topics

• [1] Derivatives of trigonometric functions
• [2] Product rule of differentiation
• [3] Chain rule of differentiation

FAQs

• Q: What is the product rule of differentiation? A: The product rule of differentiation is a rule that states that if we have a function of the form $f(x) = u(x)v(x)$, then the derivative of this function is given by $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.
• Q: How do I apply the product rule to find the derivative of a function? A: To apply the product rule, you need to identify the two functions that make up the original function. Then, you need to find the derivatives of these two functions and multiply them together. Finally, you need to add the product of the derivatives of the two functions to the product of the original functions.
• Q: What is the derivative of the function $g(x) = x \cos(x)$? A: The derivative of the function $g(x) = x \cos(x)$ is $g^{\prime}(x) = \cos(x) - x \sin(x)$.
  Q&A: Finding the Derivative of a Trigonometric Function =====================================================

Introduction

In our previous article, we explored the concept of finding the derivative of a trigonometric function. Specifically, we examined the function $g(x) = x \cos(x)$ and determined its derivative, $g^{\prime}(x)$. We used the product rule of differentiation to find the derivative of this function.

In this article, we will continue to explore the concept of finding the derivative of a trigonometric function. We will answer some common questions that students often have when it comes to finding the derivative of a trigonometric function.

Q: What is the product rule of differentiation?

A: The product rule of differentiation is a rule that states that if we have a function of the form $f(x) = u(x)v(x)$, then the derivative of this function is given by $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.

Q: How do I apply the product rule to find the derivative of a function?

A: To apply the product rule, you need to identify the two functions that make up the original function. Then, you need to find the derivatives of these two functions and multiply them together. Finally, you need to add the product of the derivatives of the two functions to the product of the original functions.

Q: What is the derivative of the function $g(x) = x \cos(x)$?

A: The derivative of the function $g(x) = x \cos(x)$ is $g^{\prime}(x) = \cos(x) - x \sin(x)$.

Q: How do I find the derivative of a trigonometric function that is not in the form $f(x) = u(x)v(x)$?

A: If the trigonometric function is not in the form $f(x) = u(x)v(x)$, then you will need to use other rules of differentiation, such as the chain rule or the power rule, to find the derivative.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation is a rule that states that if we have a function of the form $f(x) = g(h(x))$, then the derivative of this function is given by $f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$.

Q: How do I use the chain rule to find the derivative of a function?

A: To use the chain rule, you need to identify the inner function and the outer function. Then, you need to find the derivatives of these two functions and multiply them together. Finally, you need to add the product of the derivatives of the two functions to the product of the original functions.

Q: What is the power rule of differentiation?

A: The power rule of differentiation is a rule that states that if we have a function of the form $f(x) = x^n$, then the derivative of this function is given by $f^{\prime}(x) = n \cdot x^{n-1}$.

Q: How do I use the power rule to find the derivative of a function?

A: To use the power rule, you need to identify the exponent of the function. Then, you need to multiply the exponent by the function and subtract 1 from the exponent.

Conclusion

In this article, we have answered some common questions that students often have when it comes to finding the derivative of a trigonometric function. We have discussed the product rule, the chain rule, and the power rule of differentiation, and we have provided examples of how to use these rules to find the derivative of a function.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Additional Resources

• [1] Khan Academy: Derivatives
• [2] MIT OpenCourseWare: Calculus

Related Topics

• [1] Derivatives of trigonometric functions
• [2] Product rule of differentiation
• [3] Chain rule of differentiation
• [4] Power rule of differentiation

FAQs

• Q: What is the derivative of the function $g(x) = x \cos(x)$? A: The derivative of the function $g(x) = x \cos(x)$ is $g^{\prime}(x) = \cos(x) - x \sin(x)$.
• Q: How do I find the derivative of a trigonometric function that is not in the form $f(x) = u(x)v(x)$? A: If the trigonometric function is not in the form $f(x) = u(x)v(x)$, then you will need to use other rules of differentiation, such as the chain rule or the power rule, to find the derivative.
• Q: What is the chain rule of differentiation? A: The chain rule of differentiation is a rule that states that if we have a function of the form $f(x) = g(h(x))$, then the derivative of this function is given by $f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$.
• Q: How do I use the chain rule to find the derivative of a function? A: To use the chain rule, you need to identify the inner function and the outer function. Then, you need to find the derivatives of these two functions and multiply them together. Finally, you need to add the product of the derivatives of the two functions to the product of the original functions.