Derivative of xsinx

We now understand the derivative of xsinx.

It is commonly represented as d/dx(xsinx) or (xsinx)', and its value is xcosx + sinx.

The function xsinx has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Product Rule: Rule for differentiating xsinx (since it is a product of x and sinx).

Sine Function: sin(x) is a basic trigonometric function.

Cosine Function: cos(x) is another basic trigonometric function.

The derivative of xsinx can be denoted as d/dx(xsinx) or (xsinx)'.

The formula we use to differentiate xsinx is: d/dx(xsinx) = xcosx + sinx

The formula applies to all x for which the functions involved are defined.

We can derive the derivative of xsinx using proofs.

To show this, we will use the rules of differentiation and trigonometric identities.

There are several methods we use to prove this, such as:

Using the Product Rule

Using the First Principle

Using the Chain Rule

We will now demonstrate that the differentiation of xsinx results in xcosx + sinx using the above-mentioned methods:

Using the Product Rule

To prove the differentiation of xsinx using the product rule, We use the formula: xsinx = (x)(sinx)

Using the product rule formula: d/dx [u·v] = u'·v + u·v'

Let u = x and v = sinx u' = d/dx(x) = 1 v' = d/dx(sinx) = cosx

Substituting into the formula, d/dx(xsinx) = (1)(sinx) + (x)(cosx) = sinx + xcosx

Using the First Principle

The derivative of xsinx can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of xsinx using the first principle, we will consider f(x) = xsinx.

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given that f(x) = xsinx, we write f(x + h) = (x + h)sin(x + h).

Substituting these into the equation: f'(x) = limₕ→₀ [(x + h)sin(x + h) - xsinx] / h = limₕ→₀ [xsin(x + h) + hsin(x + h) - xsinx] / h = limₕ→₀ [x(sin(x + h) - sinx) + hsin(x + h)] / h = limₕ→₀ [x(cosx·h) + hsin(x + h)] / h = limₕ→₀ [xcosx + sin(x + h)]

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

Hence, proved.

Using the Chain Rule

To prove the differentiation of xsinx using the chain rule, We consider f(x) = xsinx

Let u = sinx, then f(x) = x·u Using the product rule: d/dx(x·u) = x'·u + x·u'

Substitute u = sinx and u' = cosx, d/dx(xsinx) = (1)(sinx) + (x)(cosx) = xcosx + sinx

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 0, the derivative of xsinx = xcos(0) + sin(0), which is 0. When x is π/2, the derivative of xsinx = xcos(π/2) + sin(π/2), which is 1.

• 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 differentiate the product of two functions.

• Sine Function: A basic trigonometric function represented as sinx, which describes the ratio of the opposite side to the hypotenuse in a right-angled triangle.

• Cosine Function: A basic trigonometric function represented as cosx, which describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

• Chain Rule: A rule used to differentiate composite functions.