Derivative of Si
2026-02-28 23:43 Diff
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We can derive the derivative of sin x 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:

  1. By First Principle
  2. Using Chain Rule
  3. Using Product Rule

We will now demonstrate that the differentiation of sin x results in cos x using the above-mentioned methods:

By First Principle

The derivative of sin x 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 using the first principle, we will consider f(x) = sin 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) = sin x, we write f(x + h) = sin (x + h).

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

Using limit formulas, limₕ→₀ (sin h)/h = 1 and limₕ→₀ (cos h - 1)/h = 0. f'(x) = sin x(0) + cos x(1) = cos x

Hence, proved.

Using Chain Rule

To prove the differentiation of sin x using the chain rule, We consider a composite function, but for simple sin x, it directly gives: d/dx (sin x) = cos x

Using Product Rule

We will now prove the derivative of sin x using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, sin x = (1).(sin x) Given that, u = 1 and v = sin x

Using the product rule formula: d/dx [u.v] = u'. v + u. v' u' = d/dx (1) = 0 v' = d/dx (sin x) = cos x

Again, use the product rule formula: d/dx (sin x) = u'. v + u. v'

Let’s substitute u = 1, u' = 0, v = sin x, and v' = cos x

When we simplify each term: We get, d/dx (sin x) = 0 + 1(cos x)

Thus: d/dx (sin x) = cos x.