Rivalry2

HTML Diff

0 added 0 removed

Original 2026-01-01

Modified 2026-02-28

1 We can derive the derivative of cos x using proofs.

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

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

4 By First Principle

5 Using Chain Rule

6 Using Product Rule

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

8 By First Principle

9 The derivative of cos x can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.

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

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

12 Given that f(x) = cos x, we write f(x + h) = cos (x + h).

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

14 We now use the formula for cosine<a>subtraction</a>: cos A - cos B = -2sin((A + B)/2)sin((A - B)/2). f'(x) = limₕ→₀ [ -2sin((x + h + x)/2)sin((x + h - x)/2) ] / h = limₕ→₀ [ -2sin(x + h/2)sin(h/2) ] / h = limₕ→₀ -sin(x) [ sin(h/2) / (h/2) ] · 1/2

15 Using limit formulas, limₕ→₀ (sin(h/2))/(h/2) = 1. f'(x) = -sin(x) · 1 · 1 f'(x) = -sin x

16 Hence, proved.

17 Using Chain Rule

18 To prove the differentiation of cos x using the chain rule,

19 We use the formula: cos x = 1/sin x Consider f(x) = 1 and g(x) = sin x

20 So we get, cos x = f(x)/g(x) By quotient rule: d/dx [f(x) / g(x)] = [f'(x) g(x) - f(x) g'(x)] / [g(x)]²… (1)

21 Let’s substitute f(x) = 1 and g(x) = sin x in equation (1), d/dx (cos x) = [(0) (sin x) - (1) (cos x)] / (sin x)² = -cos x / sin² x …(2)

22 Here, we use the identity sin² x + cos² x = 1.

23 Substituting this into (2), d/dx (cos x) = -sin x. Since sin x = sin x, we write: d/dx(cos x) = -sin x

24 Using Product Rule

25 We will now prove the derivative of cos x using the<a>product</a>rule.

26 The step-by-step process is demonstrated below:

27 Here, we use the formula, cos x = 1/sin x cos x = (1) · (sin x)⁻¹

28 Given that, u = 1 and v = (sin x)⁻¹

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

30 Here we use the chain rule: v = (sin x)⁻¹ = (sin x)⁻¹ (substitute v = (sin x)⁻¹) v' = -1 · (sin)⁻² · d/dx (sin x) v' = -cos x / sin² x Again, use the product rule formula: d/dx (cos x) = u'. v + u. v'

31 Let’s substitute u = 1, u' = 0, v = (sin x)⁻¹, and v' = -cos x / sin² x

32 When we simplify each<a>term</a>: We get, d/dx (cos x) = -sin x

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