Calculus • Step-by-Step

Derivative of x · sin(x) with Steps

Differentiate x·sin(x) step-by-step using the Product Rule. Detailed u and v substitutions yielding sin(x) + x·cos(x).

Problem Statement

\frac{d}{dx}[x \sin(x)]

Final Result

f'(x) = \mathrm{cos}\left(x\right) \cdot x+\mathrm{sin}\left(x\right)

Formula Required: Product Rule for Differentiation

Take the derivative of the first term times the second, plus the first times the derivative of the second.

\frac{d}{dx}[u \cdot v] = u' v + u v'

2 Steps

Set up the differentiation problem with respect to variable $x$.

\frac{d}{dx} \left[ \mathrm{sin}\left(x\right) \cdot x \right]

Apply the Product Rule to compute the derivative.

\frac{d}{dx} \left[ \mathrm{sin}\left(x\right) \cdot x \right] = \mathrm{cos}\left(x\right) \cdot x+\mathrm{sin}\left(x\right)

Interactive Calculator

Enter any equation like 2x + 8 = 20 or x² - 5x + 6 = 0 to get step-by-step solutions.

Let u = x and v = sin(x). Then u' = 1 and v' = cos(x). Product rule gives (1)(sin x) + (x)(cos x) = sin(x) + x cos(x).