Derivative of Cos 3x

We now understand the derivative of cos 3x. It is commonly represented as d/dx (cos 3x) or (cos 3x)', and its value is -3sin(3x). The function cos 3x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Cosine Function: (cos(3x) is a transformation of the basic cosine function).

Chain Rule: Rule for differentiating cos(3x) due to its composite nature.

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

The derivative of cos 3x can be denoted as d/dx (cos 3x) or (cos 3x)'.

The formula we use to differentiate cos 3x is: d/dx (cos 3x) = -3sin(3x) (or) (cos 3x)' = -3sin(3x) The formula applies to all x.

We can derive the derivative of cos 3x 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

By First Principle

The derivative of cos 3x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of cos 3x using the first principle, we will consider f(x) = cos 3x. 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 3x, we write f(x + h) = cos (3(x + h)). Substituting these into equation (1), f'(x) = limₕ→₀ [cos(3(x + h)) - cos 3x] / h = limₕ→₀ [-2sin(3x + 3h/2)sin(3h/2)] / h Using limit formulas, limₕ→₀ (sin(3h/2))/(h/2) = 3. f'(x) = -3sin(3x) Hence, proved.

Using Chain Rule

To prove the differentiation of cos 3x using the chain rule, We use the formula: Cos 3x = cos(u) where u = 3x The derivative of cos(u) is -sin(u), and the derivative of 3x is 3. By chain rule: d/dx [cos(u)] = -sin(u) * du/dx Let’s substitute u = 3x, d/dx (cos 3x) = -sin(3x) * 3 = -3sin(3x)

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(3x).

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(3x), 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 0, the derivative of cos 3x = -3sin(0), which is 0.

At points where 3x is an integer multiple of π, the derivative will also be 0 due to the sine function.

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

• Chain Rule: A rule for differentiating composite functions, which involves multiplying by the derivative of the inner function.

• Cosine Function: A trigonometric function that represents the horizontal component of an angle in the unit circle.

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

• Product Rule: A rule used to differentiate functions that are products of two other functions.