Derivative of x·sin(x)
x·sin(x) is the classic first product-rule problem: two genuinely different functions multiplied, so no shortcut exists.
Step-by-step solution
Both factors depend on x: u = x and v = sin(x). Neither is a constant, so the product rule is mandatory.
(uv)′ = u′v + uv′: differentiate one factor at a time, keeping the other frozen, then add.
u′ = 1 and v′ = cos(x), so f′(x) = 1·sin(x) + x·cos(x) = sin(x) + x cos(x).
Why it works
The two-term answer reflects two separate ways the product can change: the x factor growing while the wave holds still (contributing sin(x)), and the wave moving while x holds still (contributing x·cos(x)). The product rule is exactly this accounting — total change equals the sum of each factor's contribution with the other momentarily frozen.
The graph of x·sin(x) is a sine wave inside an ever-widening envelope of ±x, and the derivative explains its behavior: for large x the x·cos(x) term dominates, so the oscillations get steeper and steeper even though they still touch zero at every multiple of π. It's the standard example of oscillation with growing amplitude.
Common mistakes
- Multiplying the derivatives: writing 1·cos(x) = cos(x). The derivative of a product is not the product of derivatives — this is the error the product rule exists to prevent.
- Forgetting one of the two terms, usually the sin(x) from differentiating the plain x.
- Sign confusion after several steps when the problem extends to x·cos(x), whose derivative is cos(x) − x sin(x).
Practice problems
Differentiate x·cos(x)
Answer: cos(x) − x sin(x).
Differentiate x² sin(x)
Answer: 2x sin(x) + x² cos(x).
Differentiate x ex
Answer: ex(1 + x).