If y = x sinx, find $\frac{dy}{dx}$

Analyze the question: We need to differentiate y = x sin x. This is a product of two functions of x, so we use the product rule.

Concept: Product Rule If y = u·v, then

dy/dx = u·(dv/dx) + v·(du/dx)

Solution:

Let u = x and v = sin x.

• du/dx = 1
• dv/dx = cos x

Apply the product rule:

dy/dx = u(dv/dx) + v(du/dx) dy/dx = x(cos x) + sin x(1) dy/dx = x cos x + sin x

Rearranged: dy/dx = sin x + x cos x

Checking the options:

• A. sin x − x cos x: wrong sign
• B. sin x + x cos x: matches
• C. sin x − cos x: the x is missing
• D. sin x + cos x: the x is missing

Exam tip: Whenever two different types of function are multiplied (like x and sin x), think "product rule" right away. A handy way to remember it: "first times derivative of second, plus second times derivative of first." Also remember that d/dx(sin x) = cos x (positive), while d/dx(cos x) = −sin x. Mixing these up is a common source of lost marks.