If y = x sin x, find $\frac{\delta y}{\delta x}$

Analysis

The question asks for the derivative of the function $y = x \sin x$ with respect to $x$. This is a classic calculus problem involving the Product Rule, as the function is a product of two terms: $x$ and $\sin x$.

Concept: The Product Rule

When you have a function $y = u \cdot v$, where both $u$ and $v$ are functions of $x$, the derivative $\frac{dy}{dx}$ is given by the formula:

In this problem:

• Let $u = x$
• Let $v = \sin x$

Step-by-Step Solution

1. Differentiate the individual parts: • $\frac{du}{dx} = \frac{d}{dx}(x) = 1$ • $\frac{dv}{dx} = \frac{d}{dx}(\sin x) = \cos x$

2. Apply the Product Rule formula: $$\frac{dy}{dx} = (x)(\cos x) + (\sin x)(1)$$

3. Simplify the expression: $$\frac{dy}{dx} = x \cos x + \sin x$$

Rearranging the terms to match the standard format, we get:

Conclusion

Comparing our result to the given options, it matches option D.

Exam Tip

When you see a variable multiplied by a trigonometric function (like $x \sin x$, $x^2 \cos x$, or $e^x \tan x$), always remember to use the Product Rule. A common mistake students make is to differentiate both parts separately and multiply them together—avoid this! Always use the $u \frac{dv}{dx} + v \frac{du}{dx}$ structure.