Derivative of 2cos

The derivative of 2cos(x) is represented as d/dx (2cos(x)) or (2cos(x))', and its value is -2sin(x). The function 2cos(x) has a well-defined derivative, indicating it is differentiable over its entire domain. Key points include:

Cosine Function: cos(x) is a basic trigonometric function.

Derivative of Cosine: The derivative of cos(x) is -sin(x).

Coefficient Impact: The coefficient '2' scales the derivative of cos(x) by a factor of 2.

The derivative of 2cos(x) can be denoted as d/dx (2cos(x)) or (2cos(x))'.

The formula we use to differentiate 2cos(x) is: d/dx (2cos(x)) = -2sin(x)

The formula is valid for all x in the domain of the cosine function.

We can derive the derivative of 2cos(x) using several methods. To demonstrate this, we will utilize trigonometric identities along with differentiation rules:

By First Principle

The derivative of 2cos(x) can be demonstrated using the First Principle, which defines the derivative as the limit of the difference quotient. Consider f(x) = 2cos(x). Its derivative is expressed as the limit: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h

Given f(x) = 2cos(x), we write f(x + h) = 2cos(x + h).

Substituting these into the limit expression: f'(x) = limₕ→₀ [2cos(x + h) - 2cos(x)] / h = 2 limₕ→₀ [cos(x + h) - cos(x)] / h

Using the trigonometric identity: cos(x + h) - cos(x) = -2sin((2x + h)/2)sin(h/2) f'(x) = 2 limₕ→₀ [-2sin((2x + h)/2)sin(h/2)] / h = -2 limₕ→₀ [sin(h/2) / (h/2)]sin(x)

Using limit formulas, limₕ→₀ (sin(h/2) / (h/2)) = 1. f'(x) = -2sin(x) Hence, proved.

Using Chain Rule

To prove the differentiation of 2cos(x) using the chain rule, Consider f(x) = cos(x) Then, 2cos(x) = 2 * f(x) The derivative of cos(x) is -sin(x).

Using the chain rule: d/dx(2cos(x)) = 2 * d/dx(cos(x)) = 2 * (-sin(x)) = -2sin(x)

Using Product Rule

We can also verify the derivative using the product rule: Consider 2cos(x) as a product: u = 2 and v = cos(x)

Using the product rule formula: d/dx [u * v] = u' * v + u * v' u' = d/dx(2) = 0 v' = d/dx(cos(x)) = -sin(x)

Then, d/dx(2cos(x)) = 0 * cos(x) + 2 * (-sin(x)) = -2sin(x)

When a function is differentiated multiple times, the resulting derivatives are higher-order derivatives. Understanding them can be complex.

Consider a pendulum, where the initial velocity (first derivative) and acceleration (second derivative) change. Higher-order derivatives help us analyze complex oscillatory functions like 2cos(x).

For the first derivative, we write f′(x), indicating the rate of change or slope at a specific point. The second derivative, denoted as f′′(x), is derived from the first derivative. Similarly, the third derivative, f′′′(x), results from the second derivative, and this pattern continues.

For the nth Derivative of 2cos(x), we use f^(n)(x), representing the rate of change of the (n-1)th derivative.

When x = π/2, the derivative is -2sin(π/2), which is -2, since sin(π/2) = 1.

When x = π, the derivative is -2sin(π), which is 0, since sin(π) = 0.

• Derivative: The derivative of a function indicates the rate of change of the function with respect to a change in the variable.

• Cosine Function: The cosine function, written as cos(x), is a fundamental trigonometric function related to the unit circle.

• Sine Function: The sine function, written as sin(x), is another basic trigonometric function closely related to cosine.

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

• Quotient Rule: A rule used to find the derivative of a quotient of two functions.