Derivative of Cos2t

We now understand the derivative of cos(2t). It is commonly represented as d/dt (cos(2t)) or (cos(2t))', and its value is -2sin(2t). The function cos(2t) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Cosine Function: (cos(2t) is a trigonometric function).

Chain Rule: Rule for differentiating composite functions like cos(2t).

Sine Function: sin(2t) is the derivative of the cosine function, with a negative sign and a coefficient due to the chain rule application.

The derivative of cos(2t) can be denoted as d/dt (cos(2t)) or (cos(2t))'. The formula we use to differentiate cos(2t) is: d/dt (cos(2t)) = -2sin(2t) (cos(2t))' = -2sin(2t)

The formula applies to all t where the function is defined.

We can derive the derivative of cos(2t) 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
3. Using Trigonometric Identities

We will now demonstrate that the differentiation of cos(2t) results in -2sin(2t) using the above-mentioned methods:

By First Principle

The derivative of cos(2t) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of cos(2t) using the first principle, we will consider f(t) = cos(2t). Its derivative can be expressed as the following limit. f'(t) = limₕ→₀ [f(t + h) - f(t)] / h … (1) Given that f(t) = cos(2t), we write f(t + h) = cos(2(t + h)).

Substituting these into equation (1), f'(t) = limₕ→₀ [cos(2(t + h)) - cos(2t)] / h Using the trigonometric identity for cosine difference: cos A - cos B = -2 sin((A + B) / 2) sin((A - B) / 2) f'(t) = limₕ→₀ [-2 sin(2t + h) sin(h)] / h = limₕ→₀ [-2 sin(2t + h) (sin(h) / h)]

Using limit formulas, limₕ→₀ (sin h) / h = 1. f'(t) = -2 sin(2t) Hence, proved.

Using Chain Rule

To prove the differentiation of cos(2t) using the chain rule, We use the formula: Let u = 2t, thus cos(2t) = cos(u).

By the chain rule: d/dt (cos(u)) = -sin(u) du/dt d/dt (cos(2t)) = -sin(2t) (2) = -2sin(2t)

Using Trigonometric Identities

We can also prove the derivative of cos(2t) using trigonometric identities. Let u = 2t, thus: d/dt (cos(2t)) = d/dt (cos(u)) = -sin(u) (2) = -2sin(2t)

Thus, the derivative is -2sin(2t).

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(2t).

For the first derivative of a function, we write f′(t), 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′′(t). Similarly, the third derivative, f′′′(t), is the result of the second derivative, and this pattern continues.

For the nth Derivative of cos(2t), we generally use fⁿ(t) for the nth derivative of a function f(t), which tells us the change in the rate of change. (continuing for higher-order derivatives).

When t is π/2, the derivative is zero because sin(π) = 0. When t is 0, the derivative of cos(2t) = -2sin(0), which is 0.

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

• Cosine Function: The cosine function is one of the primary trigonometric functions and is written as cos(2t).

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

• Chain Rule: A rule for differentiating composite functions.

• Trigonometric Identity: An equation involving trigonometric functions that is true for every value of the occurring variables.