Derivative of Sin 4x

We now understand the derivative of sin 4x. It is commonly represented as d/dx (sin 4x) or (sin 4x)', and its value is 4cos(4x). The function sin 4x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Sine Function: (sin(x)).

Chain Rule: Rule for differentiating sin(4x) due to the multiplication of the angle by a constant.

Cosine Function: cos(x) is the derivative of sin(x).

The derivative of sin 4x can be denoted as d/dx (sin 4x) or (sin 4x)'.

The formula we use to differentiate sin 4x is: d/dx (sin 4x) = 4cos(4x)

The formula applies to all x.

We can derive the derivative of sin 4x 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 4x results in 4cos(4x) using the above-mentioned methods:

By First Principle

The derivative of sin 4x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of sin 4x using the first principle, we will consider f(x) = sin 4x. 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 4x, we write f(x + h) = sin (4(x + h)). Substituting these into equation (1), f'(x) = limₕ→₀ [sin(4(x + h)) - sin(4x)] / h = limₕ→₀ [2cos(4x + 2h) sin(2h)] / h Using the limit formula, limₕ→₀ (sin h)/h = 1, f'(x) = 4cos(4x) Hence, proved.

Using Chain Rule

To prove the differentiation of sin 4x using the chain rule, We use the formula: If y = sin u, where u = 4x, then dy/dx = dy/du * du/dx dy/dx = cos(4x) * 4 Thus, dy/dx = 4cos(4x)

Using Product Rule

Although not typically used for sine functions, the product rule can be applied in creative ways to differentiate sin 4x if the function is broken into parts. For simplicity, the chain rule remains the best approach for this function.

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(4x).

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(4x), 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 a multiple of π/4, the derivative takes specific values depending on the quadrant.

When x is 0, the derivative of sin 4x = 4cos(0), which is 4.

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

• Sine Function: A trigonometric function represented as sin(x), indicating the y-coordinate of a point on the unit circle.

• Cosine Function: A trigonometric function represented as cos(x), which is the derivative of the sine function.

• Chain Rule: A rule used in calculus for differentiating compositions of functions.

• Product Rule: A calculus rule used to find the derivative of the product of two functions.