Derivative of (cos x)^2

We now explore the derivative of (cos x)^2. It is commonly represented as d/dx ((cos x)^2) or ((cos x)^2)', and its value is -2cos(x)sin(x). The function (cos x)^2 has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below:

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

Power Rule: Rule for differentiating functions in the form of a power.

Sine Function: sin(x) is another fundamental trigonometric function.

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

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

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

We can derive the derivative of (cos x)^2 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:

1. By First Principle
2. Using Chain Rule
3. Using Product Rule

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

By First Principle

The derivative of (cos x)^2 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)^2 using the first principle, we will consider f(x) = (cos x)^2. 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)^2, we write f(x + h) = (cos (x + h))^2.

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

Using trigonometric identities and simplifying, f'(x) = limₕ→₀ [-sin(2x + 2h) + sin(2x)] / 2h As h approaches zero, this simplifies to: f'(x) = -2cos(x)sin(x)

Hence, proved.

Using Chain Rule

To prove the differentiation of (cos x)^2 using the chain rule, Let u = cos x, then (cos x)^2 = u^2. Using the chain rule, d/dx(u^2) = 2u * du/dx. The derivative of cos x is -sin x, so du/dx = -sin x.

Substituting back, we get: d/dx((cos x)^2) = 2cos(x)(-sin(x)) = -2cos(x)sin(x).

Using Product Rule

We will now prove the derivative of (cos x)^2 using the product rule. The step-by-step process is demonstrated below: Here, we express (cos x)^2 as (cos x)(cos x). Let u = cos x and v = cos x.

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

Now apply the product rule: d/dx((cos x)^2) = (cos x)(-sin x) + (cos x)(-sin x) = -2cos(x)sin(x)

Thus, the derivative is -2cos(x)sin(x).

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 x)^2.

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)^2, 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 an integer multiple of π, the derivative is zero because sin(x) is zero at these points. When x is π/2 or 3π/2, the value of the derivative is non-zero because cos(x) is zero.

• 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 that represents the adjacent side over the hypotenuse in a right triangle.

• Sine Function: A trigonometric function that represents the opposite side over the hypotenuse in a right triangle.

• Chain Rule: A formula for computing the derivative of the composition of two or more functions.

• Product Rule: A formula used to find the derivatives of products of two or more functions.