Derivative of cos2x

We now understand the derivative of cos(2x). It is commonly represented as d/dx (cos(2x)) or (cos(2x))', and its value is -2sin(2x). The function cos(2x) has a well-defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Cosine Function: (cos(x) is a trigonometric function). Chain Rule: Rule for differentiating cos(2x) as it consists of a composition of functions. Sine Function: sin(x) is the derivative of cos(x) with a negative sign.

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

We can derive the derivative of cos(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 cos(2x) results in -2sin(2x) using the above-mentioned methods: By First Principle The derivative of cos(2x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of cos(2x) using the first principle, we will consider f(x) = cos(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) = cos(2x), we write f(x + h) = cos(2(x + h)). Substituting these into equation (1), f'(x) = limₕ→₀ [cos(2(x + h)) - cos(2x)] / h = limₕ→₀ [cos(2x + 2h) - cos(2x)] / h Using the formula cos A - cos B = -2 sin((A + B)/2) sin((A - B)/2), f'(x) = limₕ→₀ [-2 sin((2x + 2h + 2x)/2) sin((2x + 2h - 2x)/2)] / h = limₕ→₀ [-2 sin(2x + h) sin(h)] / h = limₕ→₀ [-2 sin(2x + h) sin(h)/h] Using limit formulas, limₕ→₀ (sin(h)/h) = 1, f'(x) = -2 sin(2x + 0) = -2 sin(2x) Hence, proved. Using Chain Rule To prove the differentiation of cos(2x) using the chain rule, We use the formula: cos(2x) = cos(u), where u = 2x By chain rule: d/dx [cos(u)] = -sin(u) du/dx Let’s substitute u = 2x, d/dx (cos(2x)) = -sin(2x).d/dx (2x) = -sin(2x).2 Therefore, d/dx (cos(2x)) = -2sin(2x) Using Product Rule We will now prove the derivative of cos(2x) using the product rule. The step-by-step process is demonstrated below: Here, we use the formula, cos(2x) = cos(x + x) = cos(x)cos(x) - sin(x)sin(x) Given that, 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) (substitute u = cos(x)) v' = d/dx (cos(x)) = -sin(x) (substitute v = cos(x)) Again, use the product rule formula: d/dx (cos(2x)) = u'.v + u.v' Let’s substitute u = cos(x), u' = -sin(x), v = cos(x), and v' = -sin(x) When we simplify each term: We get, d/dx (cos(2x)) = -2 sin(x)cos(x) Using the identity sin(2x) = 2sin(x)cos(x), Thus: d/dx (cos(2x)) = -sin(2x) Therefore, d/dx (cos(2x)) = -2sin(x).2cos(x) = -2sin(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 cos(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 cos(2x), we generally use f(n)(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 π/2, the derivative is -2sin(π) = 0 since sin(π) is 0. When x is 0, the derivative of cos(2x) = -2sin(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: The cosine function is one of the primary six trigonometric functions and is written as cos(x). Sine Function: A trigonometric function that represents the y-coordinate of a point on the unit circle, written as sin(x). Chain Rule: A fundamental rule in calculus used to differentiate composite functions. Trigonometric Identity: Equations involving trigonometric functions that are true for every value of the occurring variables.