Derivative of cos(x/3)

We will explore the derivative of cos(x/3). It is commonly represented as d/dx [cos(x/3)] or [cos(x/3)]', and its value is -(1/3)sin(x/3). The function cos(x/3) has a well-defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Cosine Function: cos(x/3) is a cosine function with a horizontal stretch.

Chain Rule: Rule for differentiating composite functions like cos(x/3).

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

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

The formula we use to differentiate cos(x/3) is: d/dx [cos(x/3)] = -(1/3)sin(x/3)

The formula applies to all x where cos(x/3) is defined.

We can derive the derivative of cos(x/3) 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:

1. By First Principle
2. Using Chain Rule

We will now demonstrate that the differentiation of cos(x/3) results in -(1/3)sin(x/3) using the above-mentioned methods:

By First Principle

The derivative of cos(x/3) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of cos(x/3) u

sing the first principle, we will consider f(x) = cos(x/3). 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(x/3), we write f(x + h) = cos((x + h)/3).

Substituting these into equation (1), f'(x) = limₕ→₀ [cos((x + h)/3) - cos(x/3)] / h = limₕ→₀ [-2sin((2x + h)/6)sin(h/6)] / h = limₕ→₀ [-sin((2x + h)/6)sin(h/6)] / (3h/6)

Using limit formulas, as h approaches zero, sin(h/6)/(h/6) = 1. f'(x) = [-1/3]sin(x/3)

Hence, proved.

Using Chain Rule

To prove the differentiation of cos(x/3) using the chain rule, We use the formula: Cos(x/3) is a composition of functions. Consider u = x/3, then f(u) = cos(u)

The derivative of f(u) is -sin(u) and the derivative of u is 1/3.

Using the chain rule: d/dx [cos(u)] = -sin(u) * du/dx. So we get, d/dx [cos(x/3)] = -sin(x/3) * (1/3) d/dx [cos(x/3)] = -(1/3)sin(x/3)

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. T

o understand them better, think of a pendulum where the angular displacement changes (first derivative) and the rate at which this changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like cos(x/3).

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(x/3), 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 3π/2, the derivative is zero because sin(x/3) is zero there. When x is 0, the derivative of cos(x/3) = -(1/3)sin(0), which 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 trigonometric function representing the cosine of an angle, often used in waveforms and oscillations.

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

• Sine Function: A trigonometric function related to the derivative of the cosine function.

• Quotient Rule: A formula for differentiating the division of two functions.