Derivative of Sin x

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) is a fundamental trigonometric function. Basic Differentiation Rule: Rule for differentiating standard functions like sin(x). Cosine Function: cos(x) is the derivative of sin(x).

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 in the domain of sin(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: By First Principle Using Chain Rule Using Basic Differentiation 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 We use the limit formulas: limₕ→0 (sin h)/h = 1 and limₕ→0 (cos h - 1)/h = 0. f'(x) = sin x (0) + cos x (1) f'(x) = cos x Hence, proved. Using Chain Rule To prove the differentiation of sin x using the chain rule, We consider sin x as a composition of functions. Let g(x) = sin x, which is the identity function, and its derivative is cos x. Using Basic Differentiation We use the basic differentiation rule for trigonometric functions, where the derivative of sin x is directly given as 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 or any multiple of π, the derivative is 0 because cos(x) is 0 at those points. 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: The sine function is one of the primary six trigonometric functions and is written as sin x. Cosine Function: The derivative of the sine function, represented as cos x. Chain Rule: A rule for finding the derivative of a composition of functions. Quotient Rule: A method for differentiating functions that are divided by each other. ```