Derivative of xsin(x)

We now understand the derivative of xsin(x). It is commonly represented as d/dx (xsin(x)) or (xsin(x))', and can be found using the product rule. The function xsin(x) is differentiable within its domain, indicating it has a clearly defined derivative.

The key concepts are mentioned below:

Product Rule: Rule for differentiating a product of two functions, such as xsin(x).

Trigonometric Functions: Functions like sin(x), which are fundamental in calculus.

The derivative of xsin(x) can be denoted as d/dx (xsin(x)) or (xsin(x))'. The formula we use to differentiate xsin(x) is: d/dx (xsin(x)) = xcos(x) + sin(x) This formula applies to all 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.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.

To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like xsin(x).

For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.

For the nth Derivative of xsin(x), we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change, continuing for higher-order derivatives.

When x is π/2, the derivative is xcos(x) + sin(x), which simplifies to 0, as cos(π/2) = 0 and sin(π/2) = 1. When x is 0, the derivative of xsin(x) = xcos(x) + sin(x), which simplifies to 0, as both terms become 0.

• Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

• Product Rule: A rule used to find the derivative of a product of two functions.

• Trigonometric Functions: Functions like sin(x) and cos(x), fundamental in calculus.

• First Derivative: The initial result of differentiating a function, representing its rate of change.

• Second Derivative: The derivative of the derivative, indicating the change in the rate of change of a function.