Derivative of Si

We now understand the derivative of sin x. It is commonly represented as d/dx (sin x) or (sin x)', and its value is cos x. The function sin x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:

Sine Function: (sin(x) = opposite/hypotenuse in a right triangle).

Chain Rule: Rule for differentiating composite functions involving sin(x).

Cosine Function: cos(x) = adjacent/hypotenuse in a right triangle.

The derivative of sin x can be denoted as d/dx (sin x) or (sin x)'.

The formula we use to differentiate sin x is: d/dx (sin x) = cos x (or) (sin x)' = cos x

The formula applies to all x.

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.

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).

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

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

• Sine Function: A primary trigonometric function represented as sin x, indicating the ratio of the opposite side to the hypotenuse in a right-angled triangle.

• Cosine Function: A trigonometric function represented as cos x, indicating the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

• First Derivative: The initial result of a function's differentiation, showing the rate of change or slope of the function.

• Chain Rule: A rule used to differentiate composite functions by considering the derivatives of the inner and outer functions.