Derivative of Sin(x/2)

We now understand the derivative of sin(x/2).

It is commonly represented as d/dx (sin(x/2)) or (sin(x/2))', and its value is (1/2)cos(x/2).

The function sin(x/2) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Sine Function: sin(x/2) is a trigonometric function.

Chain Rule: A rule for differentiating composite functions, such as sin(x/2).

Cosine Function: cos(x/2) is another trigonometric function that appears in the derivative of sin(x/2).

The derivative of sin(x/2) can be denoted as d/dx (sin(x/2)) or (sin(x/2))'. The formula we use to differentiate sin(x/2) is: d/dx (sin(x/2)) = (1/2)cos(x/2) The formula applies to all x.

We can derive the derivative of sin(x/2) using proofs.

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

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

By First Principle

Using Chain Rule

We will now demonstrate that the differentiation of sin(x/2) results in (1/2)cos(x/2) using the above-mentioned methods:

By First Principle

The derivative of sin(x/2) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of sin(x/2) using the first principle, we will consider f(x) = sin(x/2).

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)

Given that f(x) = sin(x/2), we write f(x + h) = sin((x + h)/2).

Substituting these into equation (1), f'(x) = limₕ→₀ [sin((x + h)/2) - sin(x/2)] / h

Using the trigonometric identity for the difference of sines: sin(A) - sin(B) = 2cos((A + B)/2)sin((A - B)/2), f'(x) = limₕ→₀ [2cos((x + h)/4 + x/4)sin(h/4)] / h = limₕ→₀ [cos((x + h)/4 + x/4)·(1/2)sin(h/4)/(h/4)] · (1/2)

Using the limit formula limₕ→₀ (sin(h/4)/(h/4)) = 1, f'(x) = (1/2)cos(x/2)

Hence, proved.

Using Chain Rule

To prove the differentiation of sin(x/2) using the chain rule,

We consider the function sin(u) where u = x/2.

The chain rule states: d/dx(sin(u)) = cos(u)·du/dx

Here, u = x/2, so du/dx = 1/2. Therefore, d/dx(sin(x/2)) = cos(x/2)·(1/2)

Thus, the result is (1/2)cos(x/2).

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 sin(x/2).

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 sin(x/2), 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 = π, the derivative is (1/2)cos(π/2), which is 0 because cos(π/2) = 0. When x = 0, the derivative of sin(x/2) = (1/2)cos(0), which is 1/2 because cos(0) = 1.

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

• Sine Function: A trigonometric function represented as sin(x/2) in this context.

• Cosine Function: A trigonometric function that appears in the derivative of sin(x/2), represented as cos(x/2).

• Chain Rule: A differentiation rule used for composite functions like sin(x/2).

• Higher-Order Derivative: Refers to derivatives of a function taken successively beyond the first derivative, indicating the rate of change of previous derivatives.