Derivative of cos²(2x)

We now understand the derivative of cos²(2x). It is commonly represented as d/dx (cos²(2x)) or (cos²(2x))', and its value is -4cos(2x)sin(2x). The function cos²(2x) has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below:

Cosine Function: (cos²(2x) = (cos(2x))²).

Chain Rule: Rule for differentiating composite functions like cos²(2x).

Product Rule: A rule used when differentiating products of functions.

The derivative of cos²(2x) can be denoted as d/dx (cos²(2x)) or (cos²(2x))'.The formula we use to differentiate cos²(2x) is: d/dx (cos²(2x)) = -4cos(2x)sin(2x)

The formula applies to all x where cos(2x) ≠ 0.

We can derive the derivative of cos²(2x) 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 cos²(2x) results in -4cos(2x)sin(2x) using the above-mentioned methods:

By First Principle

The derivative of cos²(2x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of cos²(2x) using the first principle, we will consider f(x) = cos²(2x). Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = cos²(2x), we write f(x + h) = cos²(2(x + h)). Substituting these into equation (1), f'(x) = limₕ→₀ [cos²(2(x + h)) - cos²(2x)] / h Using trigonometric identities and simplifying, we use cos²A - cos²B = (cos(A + B))(cos(A - B)) f'(x) = limₕ→₀ [(cos(2x + 2h)cos(2x - 2h))] / h Upon further simplification and using limits, f'(x) = -4cos(2x)sin(2x)

Using Chain Rule

To prove the differentiation of cos²(2x) using the chain rule, We use the formula: cos²(2x) = (cos(2x))² Let u = cos(2x) Then, cos²(2x) = u² Using the chain rule, d/dx (u²) = 2u (du/dx) Here, du/dx = -2sin(2x) Substitute back, d/dx (cos²(2x)) = 2cos(2x)(-2sin(2x)) = -4cos(2x)sin(2x)

Using Product Rule

We will now prove the derivative of cos²(2x) using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, cos²(2x) = cos(2x) · cos(2x) Given that, u = cos(2x) and v = cos(2x) Using the product rule formula: d/dx [u.v] = u'.v + u.v' u' = d/dx (cos(2x)) = -2sin(2x) v' = d/dx (cos(2x)) = -2sin(2x) Using the product rule formula: d/dx (cos²(2x)) = u'.v + u.v' = (-2sin(2x))cos(2x) + cos(2x)(-2sin(2x)) = -4cos(2x)sin(2x)

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 cos²(2x).

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 cos²(2x), 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 equals π/4, the derivative is -4cos(π/2)sin(π/2) = 0 because sin(π/2) = 0.

When x equals 0, the derivative of cos²(2x) = -4cos(0)sin(0) = 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 represented as cos(x), which is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

• Chain Rule: A rule for differentiating composite functions, such as cos²(2x).

• Product Rule: A rule used when differentiating products of functions.

• Sine Function: A trigonometric function represented as sin(x), which is the ratio of the opposite side to the hypotenuse in a right-angled triangle.