Derivative of 4sinx
2026-02-28 10:04 Diff
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We can derive the derivative of 4sin 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 4sin x results in 4cos x using the above-mentioned methods:

By First Principle The derivative of 4sin x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 4sin x using the first principle, we will consider f(x) = 4sin 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) = 4sin x, we write f(x + h) = 4sin (x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [4sin(x + h) - 4sin x] / h = 4 · limₕ→₀ [sin(x + h) - sin x] / h

Using the trigonometric identity sin(A + B) = sinAcosB + cosAsinB, f'(x) = 4 · limₕ→₀ [(sin x cos h + cos x sin h) - sin x] / h = 4 · limₕ→₀ [cos x sin h + (cos h - 1) sin x] / h = 4 · [cos x · limₕ→₀ (sin h)/ h + sin x · limₕ→₀ (cos h - 1)/ h]

Using limit formulas, limₕ→₀ (sin h)/ h = 1 and limₕ→₀ (cos h - 1)/ h = 0. f'(x) = 4 · [cos x · 1 + sin x · 0] f'(x) = 4cos x. Hence, proved.

Using Chain Rule To prove the differentiation of 4sin x using the chain rule, We use the formula: 4sin x = 4 · sin x Consider y = sin x, So we get, 4sin x = 4y By the chain rule: d/dx (4y) = 4 · d/dx (y) … (1)

Let’s substitute y = sin x in equation (1), d/dx (4sin x) = 4 · d/dx (sin x) = 4cos x

Using Product Rule We will now prove the derivative of 4sin x using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, 4sin x = 4 · sin x Given that, u = 4 and v = sin x

Using the product rule formula: d/dx [u·v] = u'·v + u·v' u' = d/dx (4) = 0. (substitute u = 4) v' = d/dx (sin x) = cos x (substitute v = sin x)

Again, use the product rule formula: d/dx (4sin x) = u'·v + u·v'

Let’s substitute u = 4, u' = 0, v = sin x, and v' = cos x When we simplify each term: We get, d/dx (4sin x) = 0 · sin x + 4 · cos x Thus: d/dx (4sin x) = 4cos x