To differentiate $y = x \sin x$, we can apply the product rule, which is used when differentiating a product of two functions.

The product rule states that if $y = u(x) \cdot v(x)$, then:

$\frac{dy}{dx} = u'(x) v(x) + u(x) v'(x)$

Here, we have:

• $u(x) = x$
• $v(x) = \sin x$

First, differentiate each function:

• $u'(x) = 1$ (the derivative of $x$)
• $v'(x) = \cos x$ (the derivative of $\sin x$)

Now apply the product rule: $\frac{dy}{dx} = u'(x) v(x) + u(x) v'(x) = 1 \cdot \sin x + x \cdot \cos x$ $\frac{dy}{dx} = \sin x + x \cos x$

Looking at the options in the image, the correct answer is:

Option e. $\frac{dy}{dx} = \sin x + x \cos x$.

Would you like more details on the differentiation process or have any other questions? Here are some related questions to expand on the topic:

1. How does the product rule apply to more complex functions?
2. What is the chain rule, and when do we use it in differentiation?
3. How would we differentiate a function like $y = x^2 \sin x$?
4. Can you explain how the derivative of $\sin x$ becomes $\cos x$?
5. What is the general rule for differentiating trigonometric functions?

Tip: When differentiating, always check if you're dealing with a product, quotient, or composite functions to apply the correct rule.