Derivative of Sin(x/2)
2026-02-28 06:12 Diff
https://brightchamps.com/en-us/math/calculus/derivative-of-sin(x/2)

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).