Derivative of Cos(4x)

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

The key concepts are mentioned below:

Cosine Function: cos(x) is a periodic function.

Chain Rule: Used for differentiating composite functions like cos(4x).

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

The derivative of cos(4x) can be denoted as d/dx (cos(4x)) or (cos(4x))'. The formula we use to differentiate cos(4x) is: d/dx (cos(4x)) = -4sin(4x) The formula applies to all x.

We can derive the derivative of cos(4x) using proofs. To show this, we will use 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 cos(4x) results in -4sin(4x) using the above-mentioned methods:

By First Principle

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

Using Chain Rule

To prove the differentiation of cos(4x) using the chain rule, We use the formula: cos(4x) is a composition of cos(u) where u = 4x.

Using the chain rule:d/dx [cos(u)] = -sin(u)·du/dx Let’s substitute u = 4x, d/dx (cos(4x)) = -sin(4x)·4 d/dx (cos(4x)) = -4sin(4x) Hence, proved.

Using Product Rule

We will now prove the derivative of cos(4x) using the product rule. The step-by-step process is demonstrated below: Consider cos(4x) as a composition function: cos(4x) = cos(u), where u = 4x Using the product rule, d/dx [cos(u)] = -sin(u)·(du/dx) Substitute u = 4x, d/dx (cos(4x)) = -sin(4x)·4 d/dx (cos(4x)) = -4sin(4x) Hence, proved.

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 cos(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 cos(4x), 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 π/4, the derivative of cos(4x) = -4sin(π) = 0 because sin(π) is 0.

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

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

• Cosine Function: A periodic function that describes the ratio of the adjacent side to the hypotenuse of a right-angled triangle.

• Chain Rule: A rule used to differentiate composite functions.

• Sine Function: A trigonometric function that is the derivative of the cosine function.

• Quotient Rule: A formula used to differentiate functions that are divided by each other.