Derivative of xsin(x)
2026-02-28 06:17 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-xsin(x)

We can derive the derivative of xsin(x) using proofs. To show this, we will use the product rule along with the rules of differentiation. There are several methods we use to prove this, such as:

  1. By First Principle
  2. Using Product Rule
  3. Using Trigonometric Identities

We will now demonstrate that the differentiation of xsin(x) results in xcos(x) + sin(x) using the above-mentioned methods:

By First Principle

The derivative of xsin(x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of xsin(x) using the first principle, we will consider f(x) = xsin(x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = xsin(x), we write f(x + h) = (x + h)sin(x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [(x + h)sin(x + h) - xsin(x)] / h = limₕ→₀ [(xsin(x + h) + hsin(x + h) - xsin(x))] / h = limₕ→₀ [x(sin(x + h) - sin(x)) + hsin(x + h)] / h = limₕ→₀ [x(cos(x)h) + hsin(x + h)] / h = limₕ→₀ [xcos(x) + sin(x + h)]

Using limit formulas, limₕ→₀ sin(x + h) = sin(x). f'(x) = xcos(x) + sin(x).

Using Product Rule

To prove the differentiation of xsin(x) using the product rule, We use the formula: d/dx [u.v] = u'.v + u.v'

Here, let u = x and v = sin(x). u' = d/dx (x) = 1 v' = d/dx (sin(x)) = cos(x)

By the product rule: d/dx (xsin(x)) = (1)(sin(x)) + (x)(cos(x)) = xcos(x) + sin(x)

Using Trigonometric Identities

We can also use trigonometric identities to simplify the process. Consider xsin(x) as a product of x and sin(x), and differentiate using known identities and rules.

The derivative formula follows from the use of basic trigonometric identity simplifications and standard derivative rules.