Derivative of Cos Squared x

We now understand the derivative of cos²x. It is commonly represented as d/dx (cos²x) or (cos²x)', and its value is -2cos(x)sin(x). The function cos²x has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Cosine Function: (cos(x) is the cosine function). Chain Rule: Rule for differentiating composite functions like cos²(x). Sine Function: sin(x) is the sine function.

The derivative of cos²x can be denoted as d/dx (cos²x) or (cos²x)'. The formula we use to differentiate cos²x is: d/dx (cos²x) = -2cos(x)sin(x) (or) (cos²x)' = -2cos(x)sin(x) The formula applies to all x where cos(x) ≠ 0.

We can derive the derivative of cos²x 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: By First Principle Using Chain Rule We will now demonstrate that the differentiation of cos²x results in -2cos(x)sin(x) using the above-mentioned methods: Using Chain Rule To prove the differentiation of cos²x using the chain rule, We use the formula: Let u = cos x, then cos²x = u². The derivative of u² with respect to u is 2u. Since u = cos x, the derivative of cos x is -sin x. Therefore, d/dx (u²) = 2u(-sin x) = -2cos(x)sin(x). By First Principle The derivative of cos²x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of cos²x using the first principle, we will consider f(x) = cos²x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = cos²x, we write f(x + h) = cos²(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [cos²(x + h) - cos²x] / h = limₕ→₀ [(cos(x + h) - cos(x))(cos(x + h) + cos(x))] / h Using the trigonometric identity for the cosine difference, = limₕ→₀ [-2sin(x + h/2)sin(h/2)(cos(x + h) + cos(x))] / h This simplifies to = limₕ→₀ [-sin(h)(cos(x + h) + cos(x))] / h = limₕ→₀ (-sin(h)/h)(cos(x + h) + cos(x)) Using limit formulas, limₕ→₀ (sin h)/h = 1. f'(x) = -2cos(x)sin(x). Hence, proved.

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(x). 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(x), we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change.

When x is π/2, the derivative is 0 because sin(π/2) = 1 and cos(π/2) = 0. When x is 0, the derivative of cos²x = -2cos(0)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 primary trigonometric function, represented as cos(x), which is the adjacent side over the hypotenuse in a right-angled triangle. Sine Function: Another primary trigonometric function, represented as sin(x), which is the opposite side over the hypotenuse in a right-angled triangle. Chain Rule: A fundamental rule in calculus used to differentiate composite functions. Product Rule: A rule used to find the derivative of the product of two functions.