Derivative of sin^x

We now understand the derivative of sin^x. It is commonly represented as d/dx (sin^x) or (sin^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 basic trigonometric function).

Exponent Rule: Rule for differentiating functions raised to a power.

Chain Rule: Used for differentiating composite functions like 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) = x * sin^(x-1) * cos(x)

The formula applies to all x where sin(x) is defined.

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 Product Rule

We will now demonstrate that the differentiation of sin^x results in x*sin^(x-1)*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))^x. Substituting these into equation (1), f'(x) = limₕ→₀ [(sin(x + h))^x - (sin(x))^x] / h Using binomial expansion and limit properties, we simplify: f'(x) ≈ x*sin^(x-1)*cos(x) Hence, proved.

Using Chain Rule

To prove the differentiation of sin^x using the chain rule, We use the formula: Let u = sin(x), then sin^x = u^x Using the chain rule: d/dx [u^x] = x * u^(x-1) * du/dx Since du/dx = cos(x), d/dx (sin^x) = x*sin^(x-1)*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 = sin(x) * sin^(x-1) Given that, u = sin(x) and v = sin^(x-1) Using the product rule formula: d/dx [u*v] = u'v + uv' u' = d/dx (sin x) = cos x (substitute u = sin x) v' = (x-1) * sin^(x-2) * cos(x) (using power and chain rules) Again, use the product rule formula: d/dx (sin^x) = u'v + uv' Let’s substitute u = sin(x), u' = cos(x), v = sin^(x-1), and v' = (x-1)*sin^(x-2)*cos(x) When we simplify each term: d/dx (sin^x) = x*sin^(x-1)*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 f^n(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 an integer, the derivative's behavior might be simpler due to the power rule.

When x = 1, the derivative of sin^x = cos(x), which is the derivative of sin(x).

• 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 trigonometric functions and is written as sin(x).

• Chain Rule: A rule for differentiating composite functions, often used when a function is nested within another.

• Exponent Rule: A rule used for differentiating functions raised to a power.

• Quotient Rule: A rule for differentiating a function that is the quotient of two other functions.