Derivative of x*cos(x)
Answer
Derivative of x*cos(x)
- Rule Used
- Product Rule
- Steps
- Identify a product of two functions of x in 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: -*x+
d/dx x cos(x) = -sin(x)*x+cos(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*cos(x)
Derivative of x*cos(x) = -sin(x)*x+cos(x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-x-cos-x
Derivative
x cos(x) → -sin(x)*x+cos(x)engine: -sin(x)*x+cos(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 cos(x) uses the same product rule, but cosine's derivative is −sin(x), so one term arrives with a minus: cos(x) + x(−sin(x)) = cos(x) − x sin(x). The plus in front of cos(x) is the derivative of the factor x. Swapping the signs mixes this up with the derivative of x sin(x).
Apply the product rule to x cos(x). The form to remember, already simplified, is -sin(x)*x+cos(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 -sin(x)*x+cos(x). That number is the slope of x cos(x) there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = x cos(x).
- Apply the product rule to x cos(x). The form to remember, already simplified, is -sin(x)*x+cos(x).
- At x = 0, substitute into the simplified form -sin(x)*x+cos(x). That number is the slope of x cos(x) there; a secant over a small step in x lands next to it.
Common mistakes
Answering -x*sin(x)
−x sin(x) keeps only the cosine's contribution and drops cos(x). At x = 0 the true derivative is 1, and −x sin(x) is 0.
Differentiating only one factor of x cos(x)
Both factors depend on x. Differentiating one and copying the other, then stopping, drops a whole term.
Related rule and examples
- 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 sin(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 cos(x)?
-sin(x)*x+cos(x). x cos(x) uses the same product rule, but cosine's derivative is −sin(x), so one term arrives with a minus: cos(x) + x(−sin(x)) = cos(x) − x sin(x). The plus in front of cos(x) is the derivative of the factor x. Swapping the signs mixes this up with the derivative of x sin(x).
Which wrong answer shows up for x cos(x)?
−x sin(x) keeps only the cosine's contribution and drops cos(x). At x = 0 the true derivative is 1, and −x sin(x) is 0.
Where is the derivative of x cos(x) defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.