Derivative of x*sin(x)

Answer

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

Derivative of x*sin(x)

Rule Used: Product Rule
Steps: Identify a product of two functions of x in x* $\sin(x)$ : use the product rule (uv)' = u'v + uv'. Each factor is differentiated once while the other is held as-is, then the two terms are added.
Differentiate each factor and combine.
Final answer: $\cos(x)$ *x+ $\sin(x)$

d/dx x sin(x) = cos(x)*x+sin(x). The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.

Instant step-by-step solution for x*sin(x)

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Derivative of x*sin(x) = cos(x)*x+sin(x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-x-sin-x

Derivative

x sin(x) → cos(x)*x+sin(x)engine: cos(x)*x+sin(x)

Graph of the function and its derivative

The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.

Why the derivative has this form

x sin(x) is a product of two functions that both depend on x. Differentiate x and leave sin(x), then differentiate sin(x) and leave x: the sum is sin(x) + x cos(x). Neither term is optional. At x = 0 the second term vanishes and the derivative equals sin's value, which is 0, matching the flat start of x sin(x).

Apply the product rule to x sin(x). The form to remember, already simplified, is cos(x)*x+sin(x).

The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.

At x = 0, substitute into the simplified form cos(x)*x+sin(x). That number is the slope of x sin(x) there; a secant over a small step in x lands next to it.

Worked derivative

Write f(x) = x sin(x).
Apply the product rule to x sin(x). The form to remember, already simplified, is cos(x)*x+sin(x).
At x = 0, substitute into the simplified form cos(x)*x+sin(x). That number is the slope of x sin(x) there; a secant over a small step in x lands next to it.

Common mistakes

Answering x*cos(x)

x cos(x) differentiates only the sine and keeps x in front, dropping the sin(x) term that comes from differentiating the x.

Differentiating only one factor of x sin(x)

Both factors depend on x. Differentiating one and copying the other, then stopping, drops a whole term.

Product rule — The rule page lists this example and the others in the same family.
Derivatives of trigonometric functions — Same family of differentiation, different function.
Derivative of x cos(x) — Same family of differentiation, different function.
Derivative of x e^x — Same family of differentiation, different function.
Derivative of x ln(x) — Same family of differentiation, different function.

Frequently asked questions

What is the derivative of x sin(x)?

cos(x)*x+sin(x). x sin(x) is a product of two functions that both depend on x. Differentiate x and leave sin(x), then differentiate sin(x) and leave x: the sum is sin(x) + x cos(x). Neither term is optional. At x = 0 the second term vanishes and the derivative equals sin's value, which is 0, matching the flat start of x sin(x).